Approximate Recursive Equilibrium with Minimal State Space

Transcription

1 Approximate Recursive Equilibrium with Minimal State Space Raad, R. July 12, 2013 Department of Economics, Federal University of Minas Gerais, Brazil. Abstract This paper shows the existence of approximate recursive equilibrium with minimal state space in an environment of incomplete markets. We prove that the approximate recursive equilibrium implements an approximate sequential equilibrium which is always close to a Magill and Quinzii equilibrium without short sales for arbitrarily small errors. This implies that the competitive equilibrium can be implemented by using forecast statistics with minimal state space provided that agents will reduce errors in their estimates in the long run. We have also developed an alternative algorithm to compute the approximate recursive equilibrium with incomplete markets and heterogeneous agents through a procedure of iterating functional equations and without using the first order conditions of optimality. Keywords: Recursive Equilibrium, Computational Economics, Incomplete Markets, Approximate Equilibrium. JEL Classification: D50, D52, D58. Field: Economic Theory. 1 Introduction Magill and Quinzii (1996) provide an extension of some representative agent models such as Lucas Jr (1978) to an economy with infinite lived heterogeneous agents. They analyzed the equilibrium properties in sequential markets 1

2 under the assumption of common expectations on prices as in Radner (1972). In this model, markets are incomplete in the sense that at every date and for every commodity there will be some future dates and some events at those dates for which it will not be possible to make current contracts for future delivery contingent on those events. In this way, agents make consumption and investment plans contingent on the realizations of possible states of nature, under their own beliefs, basing the current choices on the optimal planning. The latter depends on how the agents have expectations on prices contingent on the observation of future events representing exogenous uncertainty, i.e., events whose realizations are independent of agents choices. This fact is the essence of the common expectations hypothesis which requires traders to associate the same future prices to the same future exogenous events, but does not require them to agree on the subjective probabilities associated with those events. There are two natural ways to explain how this equilibrium with common expectations can be achieved. The first regards to the high level of coordination among agents who propose to only negotiate at that contingent price level matched in the future. The second takes into account the computing capacity of the agents who use past variables and economic fundamentals to obtain a recursive relation between endogenous variables in two consecutive periods characterized, basically, by recursive transition functions. This set of functions, also called recursive equilibrium, 1 includes price estimators that represent a common recursive price expectation and implements the equilibrium in the sequential markets. Duffie et al. (1994) present a general result showing the existence of such estimators but with a domain containing a large number of variables, which calls into question the assumption that agents can compute the prices accurately. Kubler and Polemarchakis (2004), Spear (1985) and Hellwig (1982) point to a possible generic nonexistence of recursive equilibrium, with few variables in its domain, for the model of overlapping generations. But Kubler and Polemarchakis (2004) show that it is possible to obtain a stream of prices implemented recursively where agents trade goods and assets that make their utility arbitrary closed to the optimum. We can find in Citanna and Siconolfi (2010) a result of existence of recursive equilibrium, with reduced domain, outside of a residual set of 1 Sometimes we find in the literature references to recursive equilibrium as the sequential equilibrium implemented by the transition functions. Here, we find it more convenient to call the recursive equilibrium a set of transition functions, since we will show that these functions implement the sequential equilibrium. 2

3 utility functions and endowments, that is, a set of stationary utilities and endowments that is dense and is the countable intersection of open and dense sets. The methodology used in the literature to construct a recursive equilibrium is given in Duffie et al. (1994). Basically, they consider a state space S containing all pay-off relevant variables and a correspondence G : S Prob(S), where Prob(S) is the set of Borel probability measures over S. This correspondence is interpreted as the intertemporal consistency, derived from some particular model, embodying exogenous shocks, feasibility conditions, and the first order optimality conditions of dynamic programming problems of the agents. A measurable subset S S is said to be selfjustified if G(s) Prob(S ) for all s S. Under regular assumptions on G, DGMM show that, if there exists a non-empty compact self-justified set S S, then G admits a measurable selector and, using the Skorokhod s Theorem and this selector, they find a stationary function g defined 2 on S relating two consecutive realizations of the equilibrium stochastic process. The approximate recursive equilibrium defined in Kubler and Polemarchakis (2004) is a function playing the role of the selector g with domain containing few variables. However, in this case, the selector implements a competitive equilibrium which might be very far from exact equilibrium as presented in Kubler and Schmedders (2005). The latter paper exhibits an example that illustrates this fact, with zero intertemporal discount rate and the estimation error based only on the first order conditions in the optimization problem of the agents. Moreover, they develop theoretical foundations for an error analysis of approximate equilibrium providing sufficient conditions to ensure that approximate equilibrium are close to exact equilibrium. Kubler (2011) also examines the relationship between exact and approximate recursive equilibrium for economies whose fundamentals are semi-algebraic, that is, instantaneous utility and production functions can be described by finitely many polynomials. In this paper, we show the possibility of a sequential equilibrium be eventually achieved (and computed) with common expectations as in Radner (1972) or Magill and Quinzii (1996), even without assuming a high level of coordination among agents. As in Duffie et al. (1994) and in Kubler and 2 This function also depends on an extra coordinate that represents the effect of an uniform exogenous shock on the equilibrium. In some models it is possible to prove that actually, the recursive function does not depend on this process and, in this case, is called spotless recursive equilibrium. 3

4 Polemarchakis (2004), we construct estimators that implement the prices and allocations of consumption and asset shares so that agents maximize their expected utility but with arbitrarily small errors in the market clearing conditions. However, these estimators are constructed through a selector of a correspondence defined without using the first order conditions of optimality and so that one can always implement a sequential equilibrium arbitrary closed to an exact equilibrium. We do not need to assume further conditions such as in Kubler (2011) and, contrary to the definition of approximate equilibrium given in Kubler and Polemarchakis (2004), the optimality choices are exact in our approach, avoiding the criticism of Kubler and Schmedders (2005). We assume that the demand for current assets can not have arbitrarily large fluctuations as a function of asset endowments in the previous period and current prices, constructing an example with one good and one asset clarifying this assumption. Furthermore, we show that the recursive equilibrium is continuous and contains a domain with the smallest possible number of variables, also called minimal state space. 3 Contrary to Duffie et al. (1994), we use a constructive argument explaining how, in sequential markets, the equilibrium can be implemented recursively by the transition functions by showing explicitly the consecutive relations among the endogenous variables. Finally we show an alternative way to compute an equilibrium with heterogeneous agents obtaining the estimators, without using the first order conditions, through a procedure of iterating functional equations. Computing the value function and the demand through its argmax, we obtain approximations whose errors do not propagate over time as pointed out by Kubler and Schmedders (2005) about the method of Scarfh (1967) which approximates the first order conditions. Indeed, the value function is the optimal value among the current choices and all feasible plans over all future periods and the errors are generated only through the value function. 3 This set contains the portfolio asset endowment of previous period and the current state of nature. This set is minimal because an asset redistribution must change the equilibrium prices if, for instance, there exists agents with heterogeneity in the risk aversion. 4

5 2 The model 2.1 Definitions Suppose that there exists a finite set of types denoted by I = {1,..., I} and such that each type i I has a continuum of agents trading in a competitive environment. Time is indexed by t in the set N = {1, 2,...} for current periods and r N {0} for future periods. In this model, the uncertainty is exogenous, in the sense of being independent of agents actions. Each agent knows the whole set of possible exogenous variables 4 and trade contingent claims. Let be a finite set containing all states of the nature and its σ-algebra. Denote by ( τ, τ ) a copy of (, ) for all τ N. Exogenous uncertainty is described by the streams z τ = (z 1,..., z τ ) 1 τ = τ for all τ N, that is, the set of nodes of the event tree is given by τ N τ. There are a finite set J = {1,..., J} of goods and a finite set H = {1,..., H} of long lived real assets in net supply equal to one and with dividends characterized by measurable bounded functions ˆd : R JH ++ in units of goods. The matrix element ˆd jh (z) of ˆd(z) represents the amount of good j paid by one unit of asset h in the state of the nature z. Denote by Θ i R H + for all i I the convex set where asset choices are defined and C i R J + the convex set where agent i s consumption is chosen. Observe that we are not allowing for short-sales. Define the symbol without upper index as the Cartesian product. For instance, write C = i I Ci. Denote by { Q = (q c, q a ) R J + R H + : qj c + } qh a = 1 j J h H the set where the prices are defined and write Q = Q R J+H ++. The symbol q = (q c, q a ) Q stands for the consumption and asset prices in units of the numéraire respectively. We write q c ˆdh (z) := j J qc j ˆd jh (z) R + as the amount of the numéraire paid by one unit of asset h and write q c ˆd(z)θ i := h H qc ˆdh (z)θ i h R + as the amount of the numéraire paid by the portfolio θ i R H +. In addition, we define ˆd(z)θ i = h H ˆd h (z)θ i h RJ for all θ i Θ i as the amount of goods 1,..., J resulting from the ownership of the portfolio θ i. 4 Also called states of the nature or exogenous shocks. 5

6 We suppose that each asset is given in net supply one. 5 Therefore, write { Θ = θ Θ : } θ h i = 1 for all h H. i I An element θ Θ stands for the mean asset choice of the agents. Let S = Θ be the space of state variables endowed with the product topology with a typical element denoted by s = ( θ, z). Write S the Borel subsets of S and (S τ, S τ ) a copy of (S, S ) for all τ N. Write the set of all continuous 6 functions ˆq : S Q by Q and the set of all continuous functions ˆq : S Q by Q. Every Cartesian product of topological spaces is endowed with the product topology and any set of bounded continuous functions is endowed with the topology induced by the sup norm. The norm in R n considered here is the max norm, that is, x = max{ x 1,..., x n }. 2.2 Agents features In every current period, preferences are represented by a instantaneous utility given by a bounded real valued function u i : R J + R + strictly concave and strictly increasing for all i I. The symbol j u i (c i ) stands for the partial derivative of u i with respect to the j-th coordinate evaluated at the point c i. Each agent i has a measurable endowment e i : R J + of goods 1,..., J and a discount factor β i for each i I. Agents beliefs at every fixed date r are characterized by the continuous 7 map µ i r : Prob( r ) for r N, anticipating future exogenous states of the nature given the realization of the current state of nature z. We suppose that these beliefs are predictive in the context of Blackwell and Dubins (1962) with continuous probability transition rules 8 λ i : Prob(). More precisely, the measure µ i r satisfies for a rectangle A 1 A r : µ i r(z)(a 1 A r ) = λ i (z r 1, dz r ) λ i (z, dz 1 ). (1) A 1 A r 5 Since we are interested only in the case of symmetric equilibrium, we assume that each agent of the same type i chooses the same portfolio θ i and, consequently, this portfolio can be viewed as the mean asset choice of type i agents. 6 And hence bounded because S is compact. 7 The set Prob( r ) is endowed with the weak topology. 8 The space Prob() is also endowed with the weak topology. 6

7 Remark 2.1. See Stokey et al. (1989) Chapters 8 and 9 for details about the topology of Prob(), the construction of a probability measure based on the composition of probability transition rules and results about expectations over this measure. We write, with a slight abuse of notation, λ i (z)(a) as λ i (z, A) and µ i r(z)(a r ) as µ i r(z, A r ). Observe that the measure µ i r(z) is the Carathodory extension of the measure defined by the right side of (1) for rectangles. Remark 2.2. Consider a stochastic process { z τ } τ N defined on some underlying probability space and with range in the probability space. A probability transition rule on the period τ can be regarded as a generalization of the notion of conditional probability of z τ+1 given z τ with respect to agents subjective probability within the framework of Savage (1954). 9 We follow the approach of contingent choices as given in Radner (1972). As the model described here does not assume market completeness, then the agents may not be willing to make contracts that implement some future path of consumption as in the Arrow-Debreu equilibrium. Since agents do not perfectly anticipate the future states of nature, which are given exogenously, rationality leads them to make plans for the future at each current period contingent to all possible future trajectories of the states of nature. Therefore we have the definition below. Definition 2.3. The agent i s plan at some period is defined as the current period choice (c i 0, θ i 0) C i Θ i and the streams {c i r} r N and {θ i r} r N of measurable functions c i r : r C i and θ i r : r Θ i for all r N representing future plans. Remark 2.4. In each current period, the quantity c i r(z r ) can be interpreted as the value planned for consumption r periods ahead if z r is the partial history of prices actually observed at these periods. The asset plan {θ j r} r N has analogous interpretation. We assume that the agents choose a feasible plan of consumption and investment that maximizes the expected utility, under their own beliefs, among all other feasible plans. The next definitions characterize the feasibility of a plan and how agents calculate its expected value. 9 Moreover, consider ϕ i : a measurable function. If agent i assumes that the process { z τ } τ N satisfies z τ+1 = ϕ i ( z τ, ɛ τ ) for some i.i.d. error process { ɛ τ } τ N with the same distribution of ɛ, then we can define λ i (z, A) = P i (ϕ i (z, ɛ) A), where P i is the subjective probability defined for the underlying space. 7

8 Definition 2.5. Let B i : Θ i Q C i Θ i be defined as B i (θ i -, z, q) = {(c i, θ i ) C i Θ i : q c c i + q a θ i (q a + q c ˆd(z))θ i - + q c e i (z)}. Let q = {q r : r Q} r 0 be a stream of contingent prices for a given q 0 Q. A plan (c i, θ i ), for an agent i I, is feasible from (θ i -, z, q) if (c i 0, θ i 0) B i (θ i -, z, q 0 ) and (c i r(z r ), θ i r(z r )) B i (θ i r 1(z r 1 ), z r, q r (z r )) for all z r r and for all r N. Denote by F i (θ i -, z, q) the set of all feasible plans from (θ i -, z, q). Now we can define the expected utility. Definition 2.6. Let C i be the set of all sequence of measurable functions {c i r : r C i } r 0 with c i 0 C i for r N. We define the agent i s expected utility U i : C i R for consuming c i given the state z : U i (c i, z) = u i (c i 0) + (β i ) r u i (c i r(z r ))µ i r(z, dz r ). r N r Definition 2.7. Define the value function ˆv i : Θ i R by: ˆv i (θ i -, z, q) = sup{u i (c i, z) : (c i, θ i ) F i (θ i -, z, q)}. (2) The following definition characterizes the agents demand. Although this demand is independent of time, it yields the current choice at period t given some past and current observed variables for each t N. This approach allows us to use the recursive methods and compute the sequential equilibrium. Definition 2.8. We define the agent i s demand for good and asset by: 10 δ i (θ i -, z, q) = argmax{u i (c i, z) : (c i, θ i ) F i (θ i -, z, q)}. The result below can be found in Raad (2012) using that the intertemporal budget correspondence is continuous in the product topology and the Berge Maximum Theorem. 10 This correspondence can be empty when C i Θ i is not compact. 8

9 Proposition 2.9. The value function ˆv i is continuous on the set {(θ i -, z, q) : (q a 0 + q c 0 ˆd(z))θ i - + q c 0e i (z) > 0}. The lemma below ensures that the demand becomes arbitrarily large for arbitrarily low prices. Lemma For each compact C Θ Int(C Θ) there exists γ > 0 such that if ( θ, z, c, θ, q) satisfies (c i, θ i ) δ i ( θ i, z, q), (c i 0, θ i 0) C i Θ i and (c i r(z r ), θ i r(z r )) C i Θ i for all z r r and all i I then q 0 γ. Proof: See Lemma 7.13 in the Appendix. 3 Recursive and sequential equilibrium Roughly speaking, we can say that a recursive equilibrium is a function relating the variables of equilibrium between two consecutive periods. Duffie et al. (1994) shows the existence of recursive equilibrium in models with heterogeneous agents and state space containing all pay-off relevant variables. Kubler and Polemarchakis (2004) argues that a recursive equilibrium having a reduced state 11 space may not exist in a model of overlapping generations and heterogeneous agents. Furthermore, Kubler and Schmedders (2002) also shows an example of the non existence of recursive equilibrium for the case of a minimal space state pointing to the study of approximate recursive equilibrium as a solution to the problem of coordination and the calculation of the sequential equilibrium numerically. This is our goal hereafter. The next definition specifies the sequential equilibrium of the economy, also called competitive equilibrium. Definition 3.1. Consider the following measurable 1. q := {q r : r Q } r 0 contingent prices; 2. c := {c r : r C} r 0 contingent consumption allocation; 3. θ := {θ r : r Θ} r 0 contingent portfolio allocation. 11 Recall that the state space is defined as the domain of the recursive functions. 9

10 Let θ be a previous portfolio distribution and z a current state of the nature in a period t. Then these allocations and prices constitute an equilibrium for E in a period t if satisfy for all z r r : 1. optimality: (c i, θ i ) δ i ( θ i, z, q); 2. asset markets clearing: 12 i I θi r(z r ) = 1 R H ; 3. good markets clearing: i I ci r(z r ) = ˆd(z r ) 1 + i I ei (z r ). The concept approximate sequential equilibrium is similar to that given in Kubler and Polemarchakis (2004), consisting of a set of allocations of consumption and portfolio and a stream of prices balancing all markets approximately. However, contrary to Kubler and Polemarchakis (2004), here agents maximize the expected utility achieving the exact optimal benefit at that level of prices. Definition 3.2. Given ɛ > 0, consider the following measurable 1. q := {q r : r Q } r 0 contingent prices; 2. c := {c r : r C} r 0 contingent consumption allocation; 3. θ := {θ r : r Θ} r 0 contingent portfolio allocation. Let θ be a previous portfolio distribution and z a current state of the nature. Then these allocations and prices constitute an ɛ-approximate sequential equilibrium for E in some period if satisfy for all z r r : 1. (c i, θ i ) δ i (θ i -, z, q); 2. i I θi r(z r ) 1 ɛ; 3. i I ci r(z r ) e i (z r ) ˆd(z r ) 1 ɛ. We introduce now the concept of approximate recursive equilibrium and show in the appendix that it implements the sequential equilibrium of the economy. The recursive demand will be constructed using the value function. The latter is defined as the optimal value among all feasible plans, given the income and current portfolio endowments and, in addition, the transitions of the endogenous variables such as prices and asset distribution. 12 Define 1 as the vector (1,..., 1) R H. 10

11 Write Θ the space of all continuous functions ˆθ : S Θ representing the transition of asset distributions. A well known result states that for each i I there exists a bounded value function v i : Θ i S Q Θ R satisfying the Bellman Equation: } v i (θ-, i s, ˆq, ˆθ) = sup {u(c i ) + β i v i (θ i, (ˆθ(s), z ), ˆq, ˆθ)λ i (z, dz ) (3) over all 13 (c i, θ i ) B i (θ-, i z, ˆq(s)). Indeed, write V the space of all bounded continuous value functions v i : Θ i S Q Θ R endowed with the sup norm and consider the operator T i : V V, defined by } T i (v i )(θ-, i s, ˆq, ˆθ) = sup {u(c i ) + β i v i (θ i, (ˆθ(s), z ), ˆq, ˆθ)λ i (z, dz ) (4) over all (c i, θ i ) B i (θ-, i z, ˆq(s)). Clearly, T satisfies Blackwell s sufficient conditions for a contraction and hence has a fixed point. See Stokey et al. (1989) for further details. Definition 3.3. Define the agent i s consumption and portfolio policy correspondence x i : Θ i S Q Θ C i Θ i as } x i (θ-, i s, ˆq, ˆθ) = argmax {u(c i ) + β i v i (θ i, (ˆθ(s), z ), ˆq, ˆθ)λ i (z, dz ) over all (c i, θ i ) B i (θ i -, z, ˆq(s)); Remark 3.4. Notice that the policy correspondence satisfy x i (θ i -, s, ˆq, ˆθ) B i (θ i -, z, ˆq(s)) for all (θ i -, s, ˆq, ˆθ) Θ i S Q Θ. (5) Observe also that when x i is a function, then it can be written as x i = ( c i, θ i ) for continuous functions c i : Θ i S Q Θ C i and θ i : Θ i S Q Θ Θ i. The lemma below and the strict concavity of u i assure that we can assume, without loss of generality, that x i is actually a function. To show this, it suffices to note that the optimal portfolio choices are unique within the portfolio reallocations that result in the same pay off. Lemma 3.5. Suppose that C i Θ i = R J+H. Then the correspondence x i has a continuous selector. 13 Recall that s = ( θ, z). 11

12 Proof: See Lemma 7.9 in the Appendix. Definition 3.6. We say that the economy has an ɛ-approximate recursive equilibrium if there exist continuous functions ĉ i : S C i, ˆθ i : S Θ i for i I and ˆq : S Q satisfying for each s = ( θ, z) S 1. (ĉ i (s), ˆθ i (s)) x i ( θ i, s, ˆq, ˆθ) for all i I and s S; 2. i I ĉi (s) e i (z) ˆd(z) 1 ɛ; 3. i I ˆθ i (s) 1 ɛ. Example 3.7. Consider a model with one good and one asset and agents with instantaneous utility function defined by 14 u i ( ) ln( ) for i = 1, 2. Suppose that there is not exogenous uncertainty, that is, = {1} and S = Θ. Dividends are given by ˆd and there is not good endowments. We must impose that C i R ++ and Θ i R ++ because u i is defined only for R ++. The value function satisfies v i (θ-, i θ, ˆq, ˆθ) { = sup u i (c i ) + β i v i (θ i, ˆθ( θ), ˆq, ˆθ) } : (c i, θ i ) B i (θ-, i ˆq(s)) (6) and is strictly concave 15 on θ i -. Since u i is strictly concave on c i then, in this example, the policy correspondence x i given in definition 3.3 is actually function. Let θ i : Θ i Θ Q Θ Θ i and c i : Θ i Θ Q Θ C i be the policy functions as in Remark 3.4. Lemma 7.7 in appendix assures that ( c i, θ i ) is nonempty and satisfies ( c i, θ i ) > 0. Observe that: c i (θ i -, θ, ˆq, ˆθ) = ˆp( θ) θ i (θ i -, θ, ˆq, ˆθ) + (ˆp( θ) + ˆd)θ i - (7) where we write ˆp ˆq a /ˆq c. Fix (θ i -, θ, ˆq, ˆθ) Θ i Θ Q Θ with θ i - > 0. We claim that θ i (θ i -, θ, ˆq, ˆθ) = βi (ˆp( θ) + ˆd)θ i - ˆp( θ) and c i (θ i -, θ, ˆq, ˆθ) = (1 β i )(ˆp( θ) + ˆd)θ i That preferences can be viewed as a Cobb-Douglas preferences on c i and θ i by the closed form of the value function. See Faro (2013) for more details about the axiomatization of Cobb-Douglas preferences. 15 See Stokey et al. (1989) Chapter 4 for more details. 12

13 Indeed, using that u (c) = 1/c and applying the Benveniste and Scheinkman Theorem 16 given in Benveniste and Scheinkman (1979) we conclude that and hence 1 v i (θ i -, θ, ˆq, ˆθ) = u i ( c i (θ i -, θ, ˆq, ˆθ))(ˆp( θ) + ˆd) = (ˆp( θ) + ˆd)/((1 β i )(ˆp( θ) + ˆd)θ i -) = 1/((1 β i )θ i -) 1 v i ( θ i (θ i -, θ, ˆq, ˆθ), ˆθ( θ), ˆq, ˆθ) = 1/((1 β i ) θ i (θ i -, θ, ˆq, ˆθ)) for all (θ i -, θ, ˆq, ˆθ) Θ i Θ Q Θ. Therefore = ˆp( θ)/((1 β i )β i (ˆp( θ) + ˆd)θ i -) ˆp(θ) u i ( c i (θ i -, θ, ˆq, ˆθ)) = ˆp(θ)/((1 β i )(ˆp( θ) + ˆd)θ i -) = β iˆp(θ)/((1 β i )β i (ˆp( θ) + ˆd)θ i -) = β i 1 v i ( θ i (θ i -, θ, ˆq, ˆθ), ˆθ( θ), ˆq, ˆθ) which implies that θ i satisfies the first order condition on θ i of the Bellman Equation (6) replacing c i by ˆp( θ)θ i + (ˆp( θ) + ˆd)θ i -. The strict concavity of u i and of v i on the first coordinate is sufficient to conclude our assertion. The next definition provides more details of how the recursive functions can implement an ɛ-approximate sequential equilibrium. Observe that at the equilibrium each agent i has initial endowment θ i - = θ i and optimal choices on Θ, that is, each agent chooses the mean portfolio relative to its own type. Definition 3.8. We say that the functions ĉ : S C, ˆθ : S Θ and ˆq : S Q generate the process (c r, θ r, q r ) r 0 starting from θ Θ and z if for all z r r q0 = ˆq( θ, z), θ i 0 = ˆθ i ( θ, z), c i 0 = ˆθ i ( θ, z) and recursively for r N c i r(z r ) = ĉ i (θ r 1 (z r 1 ), z r ) θ i r(z r ) = ˆθ i (θ r 1 (z r 1 ), z r ) (8) for i I and q r (z r ) = ˆq(θ r 1 (z r 1 ), z r ). (9) 16 See also Stokey et al. (1989) for more details about this theorem. 13

14 The next result assures that the recursive equilibrium can actually be used to construct the sequential equilibrium. In this result, we must redefine the state space S as S = Θ because of condition 3. in Definition 3.6. Indeed, in Definition 3.8, the recursive relations must be defined in the image of ˆθ. In order to simplify notation, we keep the symbol S to denote the new state space. Theorem 3.9. If (ĉ, ˆθ, ˆq) is an ɛ-approximate recursive equilibrium then its implemented process (c, θ, q) starting from ( θ, z) S is an ɛ-approximate sequential equilibrium of the economy with initial asset holdings θ Θ and initial state of the nature z. Proof: See Theorem 7.8 in the appendix. 4 Existence and Convergence Results 4.1 Existence of an approximate recursive equilibrium In this section we will demonstrate the existence of approximate recursive equilibrium with state space S = Θ. For simplicity, we will make a slight change in notation. Notation 4.1. Write L = {1, 2,..., L} for L = H + J 2. Consider X i = C i Θ i R L + convex with 0 X and Q R L ++ the set of generalized prices. 17 Define 18 Xi as the set of all bounded continuous functions ˆx i : S X i and Q the set of all bounded continuous functions ˆq : S Q both endowed with the topology induced by the sup metric. Recall that Q is the set of all bounded continuous normalized prices ˆq : S Q. Consider V i : X i S X Q R a bounded continuous function such that V i (, s, ˆx, ˆq) is concave and increasing for all (s, ˆx, ˆq) S X Q and all i I. Moreover we suppose that V i (x i, s, ˆx, ) is homogenenous of degree zero for all (x i, s, ˆx) X i S X. Write ŵ i : S W i R J+H as the agent i s contingent endowments with W i compact convex. Recall that we consider as the max norm in R L, that is, x = max{ x 1,..., x n }. 17 In the equilibrium, actually, the prices will be normalized, that is belongs to Q Q. 18 Notice that we are using the hat symbol to denote the space of functions from S to the set specified. 14

15 The next definitions will be useful in the main result of this section. Definition 4.2. Define ˇv i : S X Q R by ˇv i (s, ˆx, ˆq) = sup { V i (x i, s, ˆx, ˆq) : x i X i and ˆq(s)x i ˆq(s)ŵ i (s) } (10) and ˇx i : S X Q R by ˇx i (s, ˆx, ˆq) = argmax { V i (x i, s, ˆx, ˆq) : x i X i and ˆq(s)x i ˆq(s)ŵ i (s) }. (11) Remark 4.3. Note that ˇv i (s, ˆx, ) and ˇx i (s, ˆx, ) are homogeneous of degree zero for all (s, ˆx) S X. We will define below the Lipschitz property. This property is related to the level of oscillation of a function. For differentiable functions it means that the function must have bounded derivative. Definition 4.4. A function f : Y R n R m is M-Lipschitz for M R ++ if f(y) f(y ) M y y for all y, y Y. When M R m ++ then we say that f = (f 1,..., f m ) is M-Lipschitz if f k : Y R n R is M k -Lipschitz for k = 1,..., m. We write Lp(M) the space of all M-Lipschitz functions. Remark 4.5. Notice that a function f : Y R n R m Lp(M) for M R m ++ then f Lp( M ). The following lemma is useful in the proof of the main result of this section. Its result will be used to construct an operator whose fixed point is the recursive equilibrium. Lemma 4.6. Consider Y R n and M, N R ++. Suppose that f : Y Y is such that f Lp(M) and g : Y Y Lp(N). Then f g Lp(MN), f + g Lp(M + N). Proof: See Lemma 7.4 in the Appendix. The next assumption holds for the model on which V i is defined using the value function v i as clarified by Theorem Assumption 4.7. For each compact X Int X there exists γ > 0 such that if ˆq Q satisfies ˆδ i (ˆx, ˆq)(S) X i for all i I and some ˆx X, then ˆq l (s) γ for all l L and all s S. 15

16 The hypothesis below precludes the possibility of a situation similar to that described by Hellwig (1982) for the case of approximate recursive equilibrium. It states that small variations in asset endowments can not produce arbitrary large fluctuations in asset demand considering that prices fluctuate in a controlled magnitude. Hellwig (1982) points to the impossibility of existence of a stationary function that relates the equilibrium variables, with rational expectations, between two arbitrary consecutive periods providing and example with demand having arbitrary large fluctuation. Remark 4.10 explains how the example 3.7 satisfies this assumption. Lemma 7.12 provides conditions in the primitives of Lucas model with heterogeneous agents so that the assumption below is satisfied. Assumption 4.8. For each compact subsets X R IL +, Q R L ++ and each i I, there exists a continuous selector of ˇx i given in (11) and denoted by ˇδ i : S X Q X i. In addition, the function ˆδ i : X Q X i defined by ˆδ i (ˆx, ˆq)(s) = ˇδ i (s, ˆx, ˆq) for all s = ( θ, z) S and all i I (12) satisfies the following property: there exists M q, M x > 0 with M x + M w M q /I and such that if ˆq Q Lp(M q ) and ˆx X Lp(M x ) then ˆδ i (ˆx, ˆq) Lp(M x ). Remark 4.9. For the case in which the selector has range in R L + we can redefine it so that the image of the new selector is contained in a given compact X i R L + and coinciding with the the initial correspondence on the inverse image of X i. Indeed, choose m i = sup X i R L ++ and define δ i (ˆx, ˆq)(s) = min{m i, ˆδ i (ˆx, ˆq)(s)} X i for all s S. Clearly, x i (s) = δ i (ˆx, ˆq) implies ˆq(s)ˆx(s) ˆq(s)ŵ i (s) and 19 δi (ˆx, ˆq) Lp(M x ) if ˆq Lp(M q ) and ˆx Lp(M x ) by Lemma 7.6 in the Appendix. Remark Clearly, the demand equation obtained in Example 3.7 satisfies the Assumption 4.8. Indeed, consider P R ++ closed. 20 Clearly, the function g i : Θ P R + defined by g i (θ-, i p) = (1 + d/p)θ- i satisfies g i Lp(M ) for some constant M, since it has bounded derivative. Consider c i and θ i as given in Example 3.7. Clearly, it is possible to find M x = (M c, M a ) and M q such that the function θ θ i ( θ i, θ, ˆq, ˆθ) = β i g i ( θ i, ˆq a ( θ)/ˆq c ( θ)) Lp(M a ) 19 See Lemma 7.6 in the appendix for details about this claim. 20 Note that P is bounded away from zero. 16

17 and that 21 c i Lp(M c ) by using (7) if ˆq Lp(M q ) and ˆθ Lp(M a ) Theorem Suppose Assumptions 4.8 and 4.7 and consider that ˆδ i defined in Assumption 4.8 for i I. Then for each fixed ɛ > 0 there exist continuous functions ˆx i : S R L + for all i I and ˆq : S Q such that ˆx i ˆδ i ɛ(ˆx, ˆq) for all i I and i I (ˆxi (s) ŵ i (s)) ɛ where is the max norm and ˆx = (ˆx i ) i I. Proof: We can suppose without loss of generality that 22 ɛ 1. Define X i a compact convex cube containing the compact set 23 X i = Pr i ({x R LI + : x i max{ŵ i (s) : s S} + 1 i I i I }) (13) in its interior relative to R L +. Write X = i I Xi and let ˆδ ɛ i the correspondence given in (12). Let ˆξ : X W R L be the excess of demand function defined by ˆξ(x, w) = i I (xi w i ). Since ˆξ is continuous and X W is compact, there exists N ξ > 0 such that ˆξ N ξ. Consider Q = {ξ R L + : ɛ 2 γ 2 /(LN ξ ) ξ N ξ }. Define Q = {ˆq : S Q ˆq Lp(M q )} and X = {ˆx : S X ˆx Lp(M x )}. Ascoli s Theorem in Royden (1963) and the compactness of Q and X assure that Q and X are compact. Define the function ˆ : X Q as ˆ (ˆx)(s) = max{ɛ 2 γ 2 /(LN ξ ), ˆξ(ˆx(s), ŵ(s))} for all s S with γ given in Assumption 4.7 with respect to the compact X. Let T : X X be the continuous convex valued correspondence (function) defined by: T (ˆx) = ˆδ i (ˆx, ˆ (ˆx)). i I Since X is a nonempty compact convex space endowed with a locally convex Hausdorff topology and T has closed graph, we can apply the Kakutani-Fan- Gliksberg Fixed Point Theorem in Aliprantis and Border (1999), Theorem 17.55, to conclude that T has a fixed point, say, ˆx. Write ˆq = ˆ (ˆx). 21 Note that ( c i, θ i ) is independent of (ĉ i, ˆθ i ). 22 In the case of ɛ > 1 remake all arguments replacing ɛ by A cube in R L is a product of a compact interval contained in R. Moreover, for a set X i R L +, write sup X i = (sup X i l ) l L R L + where X i l R + is the projection of X i into the l-th coordinate. Observe also that 0 X. 17

18 To show the market clearing conditions notice that since ˆq(s)ˆx i (s) ˆq(s)ŵ i (s) then ˆq(s)(ˆx i (s) ŵ i (s)) 0 for all s S and if we add over i I these budget restrictions we get ˆq(s)ˆξ(ˆx(s), ŵ(s)) 0 for all s S. (14) Suppose that ˆξ l (ˆx( s), ŵ( s)) > ɛγ for some s S and some l L. Write L = {k L : ˆξ k (ˆx( s), ŵ( s)) > ɛ 2 γ 2 /(LN ξ )} and L = {k L : ˆξ k (ˆx( s), ŵ( s)) ɛ 2 γ 2 /(LN ξ )}. Then ˆq k ( s) := ˆ k (ˆx)( s) = ˆξ k (ˆx( s), ŵ( s)) for all k L and 24 ˆq( s)ˆξ(ˆx( s), ŵ( s)) = k L ˆq k ( s)ˆξ k (ˆx( s), ŵ( s)) + k L ˆq k ( s)ˆξ k (ˆx( s), ŵ( s)) = ˆξ2 k (ˆx( s), ŵ( s)) + ɛ2 γ 2 ˆξk (ˆx( s), ŵ( s)) LN k L ξ k L > ɛ 2 γ 2 (L 1)N ξ ɛ 2 γ 2 /(LN ξ ) > 0 that is a contradiction with (14). We have thus proved that ˆξ l (ˆx(s), ŵ(s)) γɛ for each s S and l L. (15) To show that ˆξ(ˆx(s), ŵ(s)) ɛ 1, notice first that, using (15) we can conclude that ˆξ(ˆx(s), ŵ(s)) γɛ 1 1 for all s S and all i I. Therefore, ˆx(s) X Int X and hence ˆq(s)ˆξ(ˆx(s), ŵ(s)) = 0 for all s S. Indeed, all budget inequalities must bind because ˆξ(ˆx(s), ŵ(s)) 1 implies ˆx i (s ) Int X i for all i I. Choosing for each l L, the normalized price ˇq l (s) = ˆq l (s)/ ( k L ˆq k(s) ) for all s S, then ˆx i = ˆδ i (ˆx, ˆq) = ˆδ i (ˆx, ˇq) for all i I by Remark 4.3. Moreover, Assumption 4.7 assures that ˇq γ because ˆx(S) X Int X. Clearly, ˇq(s)ˆξ(ˆx(s), ŵ(s)) = 0 for each s S and using equation (15) we get ˇq l (s)ˆξ l (ˆx(s), ŵ(s)) = k l ˇq k (s)ˆξ k (ˆx(s), ŵ(s)) γɛ. Therefore, ˆξ l (ˆx(s), ŵ(s)) γɛ/ˇq l (s) ɛ for each l L. 24 Observe that l L. 18

19 4.2 Lucas tree model In this section we describe how to apply Theorem 4.11 to the Lucas tree model with heterogeneous agents. The Lemma 7.12 shows that Assumption 4.8 is satisfied for the case in which the model has one asset and one consumption good. Lemma Suppose that J = H = 1. Define S = {( θ, z) R+ LI θ : i I i = 1} and S its Borel σ-algebra. Write X i = C i Θ i with C i = R J + and Θ i = R H +. Suppose that V i : X i S Q X R is given by V i (c i, θ i, s, ˆq, ˆθ) = u(c i ) + β i v i (θ i, (ˆθ(s), z ), ˆq, ˆθ)λ i (z, dz ) (16) where v i is the value function given by (3). Then there exist conditions on { u i, u i, β i, λ i } i I such that Assumption 4.8 is satisfied for the demand correspondence ˇx i given by (11). Proof: See Lemma 7.12 in the Appendix. Theorem Let V i : X i S Q X R be given by (21) and ˇx i be given by (11) with respect to V i. Suppose that ˇx i satisfies Assumption 4.8 for all i I. Then Theorem 4.11 holds for {V i } i I where ˆx := (ĉ, ˆθ) and ŵ i (s) := ( ˆd(z) θ i, θ i ). Proof: In this case, all hypothesis satisfied by V i in Theorem 4.11 are assured. Indeed, Assumption 4.7 is a direct application of Lemma 2.10 and Theorem 3.9 because (ˆx, ˆq) implement the optimal choices (c, θ, q) given s = ( θ, z) such that (c i, θ i ) δ i ( θ i, z, q) for all i I. The argument used to show optimality of the equilibrium even considering C i Θ i = R J+H is standard. 25 For this, it is enough to use that u i is concave and that the optimum belongs to the interior of C i Θ i. 4.3 Convergence to an exact equilibrium In this section, we describe how the intertemporal variables of the economy could be close to a sequential competitive equilibrium through a possible 25 See Debreu (1959) or Mas-Colell et al. (1995) for further details. 19

20 anticipation of a recursive relation among them. First, we will define the concept of Jordan temporary equilibrium 26 similarly to that given in Jordan (1977) where agents can remake their plans in a given period if they fail to coordinate on the same price stream {q t } t N and show that it can be arbitrarily close to an exact sequential equilibrium with intertemporal consistency. The later would be the competitive equilibrium in which the economy would actually reach up if agents reduce their computational errors gradually. In this case, they have no incentive to deviate from their expectations on future prices and the plans do not change anymore. Notation The index tr indicates the variable planned at date t for r 1 periods ahead. In case r = 0, the index t0 indicates the current variable at period t. Definition Let θ be an initial portfolio distribution and for each period t, consider the following measurable q t (z t ) := {q tr (z t ) : r Q } r 0 contingent prices; 2. c t (z t ) := {c tr (z t ) : r C} r 0 contingent consumption allocation; 3. θ t (z t ) := {θ tr (z t ) : r Θ} r 0 contingent portfolio allocation. Then these allocations and prices constitute a Jordan temporary equilibrium for E if satisfy for all z t t and all t N: 1. optimality: (c i t(z t ), θ i t(z t )) δ i (θ i t 1,0(z t 1 ), z t, q t (z t )); 2. asset markets clearing: i I θi tr(z t )(z r ) = 1 for all r N; 3. good markets clearing: i I ci tr(z t )(z r ) = ˆd(z r ) 1+e i (z r ) for all r N. where we adopt by convention that θ i 00(z 0 ) = θ. Remark Note that if there exists t N such that the Jordan temporary equilibrium satisfies for each r N (c tr, θ tr, q tr )(z t )(z r ) = (c t+r,0, θ t+r,0, q t+r,0 )(z t+r ) then the economy eventually achieves a competitive equilibrium with intertemporal consistency, that is, a Magill-Quiinze equilibrium with infinite lived assets without short sales. 26 This concept incorporates the possibility of intertemporal inconsistency. 27 When r = 0 consider (c t0 (z t ), θ t0 (z t ), q t0 (z t )) constant. 20

21 The result below is an application of Theorem 4.11 and clarify how a competitive equilibrium can be approximated by implemented ɛ t -approximated recursive equilibrium for t N. Theorem Consider a sequence of errors {ɛ t} t N. Then there exists a sequence of ɛ t -approximate recursive equilibrium {ĉ t, ˆθ t, ˆq t } t N with ɛ t ɛ t and a Magill-Quiinze competitive equilibrium {c tr, θ tr, q t } r 0 with initial distribution θ and such that the distance 28 from the equilibrium implemented by {ĉ t, ˆθ t, ˆq t } t N starting from (θ t0 (z t 1 ), z t ) and {c tr, θ tr, q tr } r 0 is smaller than ɛ t for all t N. Proof: Fix t and suppose that s t = (θ t 1,0 (z t 1 ), z t ) is given. Choose ɛ k = 1/k for k N. Consider {ĉ k, ˆθ k, ˆq k } the ɛ k -approximate recursive equilibrium and {c trk, θ trk, q tk } r 0 its implemented ɛ k -approximate sequential equilibrium choices and plans starting from s t. Since is finite, 29 we can choose, passing to a subsequence, {c trkn, θ trkn, q tkn } n N such that {c tr, θ tr, q t } := lim n {c trkn, θ trkn, q tkn } for all r N. We claim that the limit prices {q t } t N are positive. Indeed, the compact set X given in 4.11 was chosen independent of ɛ and hence, Lemma 2.10 can be applied. Using Proposition 2.9 we conclude that the value function ˆv j given in Definition 2.7 is continuous on (θt 1,0(z j t 1 ), z t, q t ) and hence (c j t, θ j t ) δ j (θt 1,0(z j t 1 ), z t, q t ). Therefore the profile {c t, θ t, q t } t N is actually a Magill-Quiinze equilibrium with infinite lived assets without short sales because plans and choices satisfy the exact market clearing conditions, by construction and equations (18) and (19), defining the optimal intertemporal consistent plans {c tr } r 0 and {θ tr } r 0 as the limit of implemented plans. To conclude the proof, for a fixed t N, pick n N such that ɛ n := 1/k n ɛ t and that d((c tkn, θ tkn, q tkn ), (c t, θ t, q t )) ɛ t. Choosing ɛ t = ɛ n and (ĉ t, ˆθ t, ˆq t ) = (ĉ kn, ˆθ kn, ˆq kn ), we get the desired property using induction on t with initial conditions {s t } t N. 28 See Aliprantis and Border (1999) Section 3.9 about the metric in the Cartesian product. 29 Using the product topology, the Tychonoff Product Theorem and the fact that {c trk, θ trk, q trk } k N is contained in a compact set. 21

22 Remark Theorem 4.17 allow us to conclude that if agents update the recursive statistics with arbitrary small errors then, in the long run, they are close to the behavior of competitive equilibrium models presented in Radner (1972) or Magill and Quinzii (1996) with intertemporal consistency, even basing their choices in anticipation of approximate future prices using continuous statistics with minimal state space to compute the equilibrium. We conclude that the transition functions of the recursive equilibrium represent an important instrument of coordination among agents if the measurement errors become small enough to avoid that agents deviate from their previous plans. 5 Numerical Method The numerical method used here is similar to that found in Judd (1998). Basically, we proceed iterating functions recursively to find the solution of a certain functional equation as a fixed function on the limit 30. To clarify the method, given a price ˆq and a transition ˆθ, write ˆθ ( θ i, θ, z, ˆq, ˆθ) = ( θ i ( θ i, θ, z, ˆq, ˆθ)) i I for all ( θ, z) S where θ i : Θ i Θ Q Θ Θ i is the argmax of Bellman Equation (3). We compute first the value function v i (,, ˆq, ˆθ) on a finite set called grids S using the standard Bellman Method iterating value functions given an initial function v i 0. After that, we compute the asset excess demand ˆξ a ( θ, z) = i I θ i ( θ i, θ, z, ˆq, ˆθ) 1 for all ( θ, z) S, using the argmax of the value function v i (,, ˆq, ˆθ) on grids. Choosing a price ˆq = ˆq + ˆq Q where ˆq is an increment proportional to excess demand function ˆξ a, with the proportionality constant chosen appropriately to ensure the speed of convergence; we compute again ˆθ = { θ i ( θ i, θ, z, ˆθ, ˆq )} i I for all ( θ, z) grids and repeat the previous step until the desired precision. Notice that the convergence of the algorithm implies that the increment ˆq is near zero and therefore the excess demand ˆξ a also tends to zero in the limit. 30 On the sup norm. 22

23 Figure 1: Graphics of θ 1 ˆθ 1 ( θ 1, 1 θ 1 ) and θ 2 ˆθ 2 (1 θ 2, θ 2 ) for e 1 (z) = p1 and λ 2 (z) = p2 for all s {z 1, z 2 } Figure 2: Graphics of θ 1 ˆq( θ 1, 1 θ 1, z 1 ) and θ 1 ˆq( θ 1, 1 θ 1, z 2 ) We suppose that agents have homogeneous beliefs λ i = λ and heterogeneous instantaneous income e i : R ++ depending on the states of the nature. The budget set becomes for i = 1, 2 B i (θ i -, z, q) = {(c i, θ i ) C i Θ i : q c c i + q a θ i (q a + q c ˆd(z))θ i - + q c e i (z)} and we choose e 1 (z 1 ) = 1, e 1 (z 2 ) = 1, e 2 (z 1 ) = 1 and e 2 (z 2 ) = 2 that is, this example includes aggregate risk on the income. Type 1 has initial asset endowment θ 1 0 = 0.1 and type 2 has initial asset endowment θ 1 0 = 0.9. As a result we get linear affine policy functions and the dynamics of the simulations shows that all agents survive and the consumption and asset stream eventually becomes a stationary process. Figure 1 shows the policy functions. It is clear graphically that these functions are linear affine. Figure 2 shows that prices are independent of the income distribution and figure 3 shows that average consumption and asset stream becomes stationary over time. 6 Conclusion There exists a recursive equilibrium with minimal state space implementing a sequential equilibrium arbitrarily closed to a Magill Quinzii competitive equilibrium. Therefore, if agents update the recursive statistics with arbitrary small errors then, in the long run, they are close to the behavior of competitive equilibrium models presented in Radner (1972) or Magill and Quinzii Figure 3: Graphics of the asset average stream r 5000 (ci 1t)(z t r)/5000 for i = 1, 2 on the left, r 5000 (θi 2t)(z t r)/5000 for i = 1, 2 on the right, r = 5000 trajectories of (z r1, z r2,..., z rt ), T = 100 and λ 2 (z) = p = (0.4, 0.6) for all s {z 1, z 2 }. 23

24 (1996) with intertemporal consistency, even basing their choices in anticipation of approximate future prices using continuous statistics with minimal state space to compute the equilibrium. We conclude that the transition functions of the recursive equilibrium represent an important instrument of coordination among agents if the measurement errors become small enough to avoid that agents deviate from their previous plans. Moreover, it is possible o compute the approximate recursive equilibrium without using the first order conditions of optimality, through a method of iterating functional equations. 7 Appendix Notation 7.1. Recall that L = {1, 2,..., L}. Consider, X i R L +, Q = {q R L + : l L q l = 1}, Q = Q R L ++, Q γ = Q [γ, 1] L and S a topological space. Define X i as the set of all bounded continuous functions ˆx i : S X i and Q the set of all bounded continuous functions ˆq : S Q both endowed with the topology induced by the sup metric. Define Q analogously. The absence of the upper index stands for the Cartesian product on i I and we write 1 = (1, 1,..., 1) R L +. Lemma 7.2. Suppose that X i R L + is a compact convex set with 0 X i and that W i = R L +. Let B i : W i Q X i be the budget correspondence defined by B i (w i, q) = {x i X i : qx i qw i }. Then B i is continuous. Proof: If X i = {0} is trivial. Suppose that X i {0}. The upper hemicontinuity follows from the fact that B i has closed graph and compact range space. To show the lower hemicontinuity, let (w i n, q n ) A i converging to ( w i, q) A i as n and x i B i ( w i, q). Suppose first that q w i > 0. Then there exists an open set 31 O of A i containing ( w i, q) such that qw i > 0 for all (w i, q) O. Let Int B i : O X i be the correspondence defined by Int B i (w i, q) = {x i X i : qx i < qw i }. 31 Recall that we are using the relative topology. 24

25 Since 0 X i, Int B i is nonempty on the set O and X i is convex, then 32 B i (w i, q) = cl[int B i (w i, q)] for all (w i, q) O. Clearly, Int B i has open graph. Therefore, using that an open graph correspondence is lower hemicontinuous and that the closure of a lower hemicontinuous correspondence is lower hemicontinuous, we conclude that B i is lower hemicontinuous on O and hence there exists an N N and a sequence 33 x i n B i (wn, i q n ) for each n N such that x i n x i as n. If q w i = 0 then x i = 0. Since q 1 > 0, 0 X i and X i is convex non degenerated, there exists N N such that q 1n > 0 and (q n wn/q i 1n, 0) X i for n N. Choose the sequence x i 1n = q n wn/q i 1n and x i ln = 0 for l > 1 and n N. Then x i 1n = q n wn/q i 1n q w i / q 1 = 0 and hence x i n x i = 0 as n. Moreover, by construction, x i n B i (wn, i q n ) for each n N. Lemma 7.3. Consider Ỹ, Y metric spaces with Ỹ compact and Ỹ Y endowed with the product topology. Suppose that f : Ỹ Y RL is continuous. Let C(Ỹ, RL ) be the space of all bounded continuous functions h : Ỹ RL endowed with the sup norm. Then the function g : Y C(Ỹ, RL ) defined by g(y)( ) = f(, y) is continuous. Proof: Consider a sequence y n y and fix ɛ > 0. Then the set Y = {y n } n N {y} is compact. Therefore, f is uniformly continuous on Ỹ Y and hence the exists δ > 0 such that for all ỹ, ỹ Ỹ d((ỹ, y ), (ỹ, y)) < δ and y Y implies f(ỹ, y ) f(ỹ, y) ɛ. Assuming that 34 d((ỹ, y ), (ỹ, y)) = dỹ (ỹ, ỹ) + d Y (y, y) then choosing n 0 such that n n 0 implies d Y (y n, y) < δ we conclude that n n 0 implies d((ỹ, y n ), (ỹ, y)) < δ for all y Ỹ and hence f(ỹ, y n) f(ỹ, y) ɛ for all ỹ Ỹ. Thus f(, y n) converges uniformly to f(, y). Lemma 7.4. Consider Y R n. Suppose that f : Y Y and g : Y Y satisfy f Lp(M f ) and g Lp(Mg). Then f g Lp(M f M g ). Moreover, 32 To see the inclusion B i (w i, q) cl[int B i (w i, q)], given x i B i (w i, q) notice that if we choose x i Int B i (w i, q) then x i n := (1 1/n)x i + x i /n Int B i (w i, q) and x i n x i. Thus x i cl[int B i (w i, q)]. 33 The set N is chosen such that (w i n, q n ) O for each n N. 34 Clearly, this metric induces the product topology on Ỹ Y. 25

26 if f N f and γ g g N g for n = 1 and N f, N g, γ g > 0 then h = f/g Lp(N g M f + N f M g )/γg), 2 the sum f + g Lp(M f + M g ) and the product fg Lp(N g M f + N f M g ). Proof: Fix y, y Y. Thus f(g(y)) f(g(y )) M f g(y) g(y ) M f M g y y. Moreover, for n = 1 Therefore, Furthermore, f(y)g(y ) f(y )g(y) f(y)g(y ) f(y )g(y ) + f(y )g(y ) f(y )g(y) g(y ) f(y) f(y ) + f(y ) g(y) g(y ) (N g M f + N f M g ) y y. h(y) h(y ) = f(y)g(y ) f(y )g(y) / g(y)g(y ) y y (N g M f + N f M g )/ g(y)g(y ) (N g M f + N f M g ) y y /γ 2 g. f(y)g(y) f(y )g(y ) f(y)g(y) f(y )g(y) + f(y )g(y) f(y )g(y ) g(y) f(y) f(y ) + f(y ) g(y) g(y ) (N g M f + N f M g ) y y. Lemma 7.5. Consider Y, Y, Y k R n : k = 1, 2. Suppose that f : Y 1 Y 2 Y and g k : Y Y k satisfy f Lp(M f ) and g k Lp(M k ) for k = 1, 2. Then h : Y Y defined by h(y) = f(g 1 (y), g 2 (y)) for all y Y satisfies h Lp(M f max{m 1, M 2 }). Proof: h(y) h(y ) = f(g 1 (y), g 2 (y)) f(g 1 (y ), g 2 (y )) M f max{ g 1 (y) g 1 (y ), g 2 (y) g 2 (y ) } M f max{m 1 y y, M 2 y y } = M f max{m 1, M 2 } y y 26