On the Interaction between Risk Sharing and Capital Accumulation in a Stochastic OLG Model with Production ?

Martin Barbie Department of Economics, University of Karlsruhe, Kollegium IV am Schloss, D-76128 Karlsruhe, Germany

Marcus Hagedorn European Central Bank, Kaiserstr. 29, 60311 Frankfurt, Germany

Ashok Kaul ∗ Gutenberg University Mainz, Department of Economics - FB 03, D-55099 Mainz, Germany

? We are indebted to Subir Chattopadhyay, Piero Gottardi, Georg N¨oldeke, Herakles Polemarchakis, and Itzhak Zilcha for helpful comments and to Don Brown, and Gabrielle Demange for helpful discussions. We would like to thank an anonymous associate editor and two anonymous referees for many detailed and very helpful suggestions. ∗ Corresponding author. Email addresses: [email protected] (Martin Barbie), [email protected] (Marcus Hagedorn), [email protected] (Ashok Kaul).

Preprint submitted to Elsevier

5 February 2007

Abstract We analyze the interaction between risk sharing and capital accumulation in a stochastic OLG model with production. We give a complete characterization of interim Pareto optimal competitive equilibrium allocations. Furthermore, we provide tests of Pareto optimality/suboptimality based on (risky) rates of return only. Key words: Stochastic OLG Model, Risk Sharing, Dynamic Efficiency, Pareto Optimality JEL classification: D61, D91, E13, E43

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1

Introduction

In this paper, we analyze the interaction between optimal intergenerational exchange under uncertainty (”risk sharing”) and capital accumulation in a stochastic OLG model with production. Our contribution is twofold. First, we characterize Pareto optimality with production. Second, we examine the relation to dynamic efficiency and deal with observational implications of theoretical conditions of (sub-)optimality. As is well known, in pure exchange OLG economies the first welfare theorem fails to hold, i.e., competitive equilibria may fail to be Pareto optimal [Allais [1], Samuelson [21]]. A characterization of Pareto optimality in a pure exchange OLG model under certainty was given by Balasko and Shell [2] and and Okuno and Zilcha [18]. Under uncertainty, a characterization of (interim) Pareto optimality in a pure exchange OLG model was derived by Chattopadhyay and Gottardi [9]. Another strand of literature in capital theory with a long-standing tradition is concerned with an a priori stronger form of inefficiency that is independent of specific household preferences, namely the literature on the possibility of capital overaccumulation (dynamic inefficiency) [see Bertocchi [3,4] for a good discussion]. Of course, saving too much capital in a competitive equilibrium in the sense that an alternative capital path can uniformly increase aggregate consumption implies Pareto suboptimality. This literature was initiated by Malinvaud [17] and then extended to growth models by Phelps [20] and Diamond [13] and more general infinite-horizon production problems by Cass [7]. 1 An extension of the dynamic efficiency issue under certainty to a setting 1

Tirole [22] analyzed the relationship between dynamic efficiency and the existence

of bubbles as well as the Pareto optimality of bubbly equilibria.

3

with uncertainty has been given by Zilcha [23] and Dechert and Yamamoto [10]. Characterizations in terms of prices in competitive equilibrium of these two notions of efficiency - although the notions themselves are conceptually distinct - are based on the same formal argument, so that the conditions in terms of competitive equilibrium prices for dynamic inefficiency and Pareto optimality in pure exchange models coincide under certainty [see Cass [7] and Okuno and Zilcha [18] or Balasko and Shell [2]]. Under uncertainty, risk sharing issues make important adaptations and qualifications necessary. Chattopadhyay and Gottardi [9] extend and generalize a Cass-type argument to a stochastic pure exchange model. Our first contribution (Proposition 1) is a complete characterization of interim Pareto optimality in a stochastic Diamond model, extending the proof by Chattopadhyay and Gottardi [9] to a setup with production. 2 It turns out that in a competitive equilibrium the characterization of interim Pareto optimality is equivalent to the one in a pure exchange setup. This holds although in a model with production there are more feasible deviations from the competitive equilibrium than under pure exchange and it is not a priori possible to restrict attention to pure transfer schemes. As a special case of our optimality characterization we obtain Zilcha’s [23] dynamic efficiency characterization (Proposition 2). Contrary to the case of certainty, the conditions for dynamic efficiency and Pareto optimality do not 2

Demange and Laroque [12] derive a partial classification (but not a characteri-

zation) of interim Pareto optimality in a stochastic OLG model with production, however under relatively strong stationarity assumptions.

4

coincide under uncertainty. This means that the possibility of overaccumulation of capital is not necessarily related to what Manuelli [15] called the ”intergenerational risk sharing” part of the efficiency problem. Nevertheless, we show that there is a relationship between the risky rate of return on capital - and more generally the rate of return of some arbitrary asset - and interim Pareto optimality of competitive equilibria (Proposition 3). Using this relation, we derive a sufficient condition for interim Pareto optimality that requires only the knowledge of the rates of return of some arbitrary asset for each date-event and not the full set of contingent claims prices. This set of results is related to recent independent research by Chattopadhyay [8] but is more general [see the discussion after Proposition 4]. The paper is organized as follows. Section 2 describes the stochastic Diamond model. Section 3 gives a complete characterization of interim Pareto optimality in a competitive equilibrium. Section 4 reconsiders the condition for dynamic efficiency and derives optimality tests based on risky and other rates of return. All proofs are given in the appendix.

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The Model

We consider a stochastic version of Diamond [13]. Uncertainty enters via shocks to the production technology. Time is discrete, starts at 0 and extends infinitely into the future. There is production and a consumption-savings decision at every point of time. Production The production technology at time t is described by a neoclassical constant-returns-to-scale production function F : R2+ × St → R+ , where 5

F (Kt , Lt , θt ) is the output produced at time t, Kt is the capital stock, Lt is the labor input and θt is the current stochastic shock. The perishable good produced by the technology is the only good in the economy and is used for production and consumption. For simplicity, the depreciation rate is assumed to be 1. For convenience we work with the per-capita production function f (kt , θt ) = F



Kt , 1, θt Lt



erties: f (0, θt ) = 0, f 0 =

(where k = K/L) with the following standard prop∂f ∂k

> 0, f 00 < 0, f 0 (0, θt ) = ∞, f 0 (∞, θt ) = 0.

Uncertainty W.l.o.g, for all t ≥ 1, the set of production shocks, St = S, has finite cardinality where |S| is the cardinality of S and S0 is single-valued. There is a system of strictly positive transition probabilities qt+1 (θt+1 | σ) that are contained in a compact subset of the interior of the |S| − 1 dimensional unit simplex (where σ := (θ0 , θ1 , θ2 , ..., θt )). Since the production shocks are the only source of uncertainty in the economy, it is possible to describe the uncertainty by a date-event tree, where σ0 = {θ0 } is the root and σ = (θ0 , θ1 , θ2 , ..., θt ) is a generic node. If we want to stress the date of a node we write σt . The set of nodes at time t is therefore S0 ×S1 ×S2 ×...×St and denoted by Σt . The date-event tree is denoted by Γ and in slight abuse of notation we will identify it with the set of its nodes ∪t≥0 Σt . Every node σ = (θ0 , θ1 , θ2 , ..., θt ) has a unique immediate predecessor, denoted σ−1 , which is equal to (θ0 , θ1 , θ2 , ..., θt−1 ). σ − denotes the set of all predecessor nodes of node σ. σ + denotes the set of nodes which are immediate successors of node σ, i.e. the set of all nodes for which σ is the immediate predecessor. A path is a sequence of nodes {σt }t≥0 such that σt+1 ∈ σt+ . A generic path will be denoted by σ ∞ and can be identified with a unique sequence of shocks (θ0 , θ1 , θ2 , ...) from the set Σ := S0 × S1 × S2 × .... The t-th coordinate of the path σ ∞ is denoted by σt∞ . t (σ) denotes the point of time at which event 6

σ ∈ Σt occurs. We also identify a date-event tree Γ with the paths that it e contains, i.e. we use the obvious isomorphism between Σ and Γ. Any subset Γ e has an immediate successor node σ e b ∈Γ of Γ is called a subtree if each σ ∈ Γ e such that each σ ∈ Γ, e σ 6= σ e 0 , is a successor and there exists a unique σe0 ∈ Γ

node of σe0 . Furthermore, denote by Γ (σe ) the (full) subtree of Γ that has σe ∈ Γ as its root and includes all successor nodes of σe . Households For simplicity we assume that there is no population growth, Lt = L = 1 for all t. At each node in the tree, one household is born who lives for two periods. Hence households are distinguished according to the date and the state of nature in which they are born. Since time starts at t = 0, we have one initially old household (born in period −1). This household has preferences that are strictly monotone in the single consumption good in period 0. His consumption in period 0 is denoted by co (σ0 ). All other households face uncertainty during their second period of life due to 1+|S|

the production shocks. The consumption set of a household born in σ is R+

.

His von Neumann-Morgenstern preferences are described by a function Eσ P

u (c (σ)) := 

θt+1 ∈S

qt+1 (θt+1 | σ) · u (cy (σ) , co ((σ, θt+1 ))) . Here we denote by 

c (σ) = cy (σ) , (co (σ 0 ))σ0 ∈σ+ the consumption vector of household σ, cy (σ) is his consumption in his youth, (co (σ 0 ))σ0 ∈σ+ is his consumption in the different states of nature in his old age. The (Bernoulli) function u : R2+ → R is twice continuously differentiable (in the interior of its domain), strictly increasing in each argument and with negative definite Hessian matrix. In his youth, each household receives a wage income w (σ) from inelastically supplying one unit of labor. Facing state contingent prices p (σ) and p (σ 0 ), σ 0 ∈ 7

σ + , the household’s problem is to maximize utility subject to his intertemporal budget constraint:

max Eσ u (c (σ)) (cy (σ),(co (σ0 ))σ0 ∈σ+ )∈R1+|S| +

s.t. p (σ) · cy (σ) +

X

(1)

p (σ 0 ) · co (σ 0 ) = p (σ) · w (σ) .

σ 0 ∈σ +

Since only one individual is born per node, markets are complete once an individual is born. With several (heterogeneous) households born per node, we would simply introduce a complete set of Arrow securities explicitly. Firm The firm’s problem is to decide after the shock realization at each node σ how much capital to invest. This capital is then used to produce output at the successor nodes of σ. The firm maximizes expected profits given Arrow Debreu prices p (σ) . Let k (σ) be the firm’s investment in state σ. The firm’s problem at a node σ at time t is: 3

max

k(σ)≥0

X

[p (σ 0 ) · f (k (σ) , θt+1 )] − p (σ) · k (σ) .

(2)

σ 0 ∈σ +

Next, we define feasible allocations, the notion of optimality and a competitive equilibrium. Our definition of Pareto optimality was introduced by Muench [16] and Peled [19]. It is also used in Chattopadhyay and Gottardi [9]. Households born in different date-events are considered as distinct households. Then, 3

Problem (2) describes the problem of finding the optimal capital-labor ratio.

The original problem of the firm is to choose optimal capital and labor. Since the production function has constant returns to scale, in equilibrium only the optimal capital-labor ratio is determined, the firm is indifferent about absolute quantities.

8

the usual concept of Pareto optimality is applied to this set of households. Following the terminology of Demange and Laroque [11] we call this interim Pareto optimality. This feature of our model distinguishes our analysis e.g. from Blanchard and Weil [5] or Gottardi and Kubler [14] who adopt ex ante Pareto optimality as a welfare criterion. Together with the fact that markets are complete once an individual is born we may conclude that sequentially complete markets are an important maintained feature of our setup throughout the paper.

Definition 1 Let the initial capital stock k−1 be given. A feasible allocation is a tuple (c, k) = (co (σ0 ) ,



cy (σ) , (co (σ 0 ))σ0 ∈σ+





σ∈Γ



), (k (σ))σ∈Γ , k−1 ) such

that (1) co (σ0 ) + cy (σ0 ) + k (σ0 ) = f (k−1 , θ0 ), (2) For σ ∈ Γ : co (σ 0 ) + cy (σ 0 ) + k (σ 0 ) = f (k (σ) , θ)

∀σ 0 = (σ, θ) ∈ σ + .

Definition 2 A feasible allocation (c, k) is called interim Pareto optimal if 



there exists no other feasible allocation cb, kb such that cbo (σ0 ) ≥ co (σ0 ) and Eσ u (cb (σ)) ≥ Eσ u (c (σ)) for all σ ∈ Γ, with at least one strict inequality. Definition 3 (c∗ , k ∗ , p∗ , w∗ ) is a competitive equilibrium if c∗ solves the household’s problem (1) for every household given competitive wages w∗ and prices p∗ , firms maximize profits given p∗ , and (c∗ , k ∗ ) is a feasible allocation, i.e., markets clear.

Throughout the rest of the paper, we say that a competitive equilibrium (c, k) or more generally an allocation (c, k) is interior if the capital stock k is uniformly bounded away from 0 (given the initial capital stock k−1 ). 9

3

Interim Pareto Optimality

We start with the definition of a transfer pattern. Chattopadhyay and Gottardi [9] introduced an equivalent concept in their important paper. Definition 4 A transfer pattern λ is a function λ : Γ → [0, 1] with the fol· S

· S

lowing properties. Γ can be partitioned, Γ = Γ+ Γ0 {σ0 } with Γ+ 6= ∅, so that the following holds: (1) λ (σ0 ) = 1. (2) σ ∈ Γ0 if and only if λ (σ) = 0 and σ ∈ Γ0 implies for all σ 0 ∈ σ + that σ 0 ∈ Γ0 . (3) If σ ∈ Γ+ then

P

σ 0 ∈σ +

λ (σ 0 ) = 1.

The definition says in detail that 1. Γ can be divided into three disjoint subsets, one set of nodes Γ0 where no transfers are assigned (the value of λ is set to zero at such nodes), one set of nodes Γ+ associated with positive transfers, and the root σ0 . As a convenient convention transfer weight one is assigned to the root. 2. If no transfers are assigned at one node then no transfers are assigned at all successor nodes of this node. 3. Most importantly, once transfers are assigned at one node σ then they are assigned at some direct successor node(s) σ 0 ∈ σ + . Furthermore, transfers are normalized so that their weights sum up to one. Now we can state our characterization result. Note that under our assumptions we can derive the well-known uniform Gaussian curvature conditions from primitives and do not have to assume them (see the appendix). Proposition 1 An interior competitive equilibrium allocation is not interim Pareto optimal if and only if there exists a transfer pattern λ and a finite 10

positive number B such that for every path σ ∞ =(σ0∞ ,σ1∞ , ...) in the tree ∞ X

t 1 Y λ (σs∞ ) ≤ B. ∞ ) p (σ t t=1 s=1

(3)

Remark 1 This condition coincides with the conditions for interim Pareto (sub-)optimality in a pure exchange stochastic OLG model derived by Chattopadhyay and Gottardi [9].

To prove that the condition in Proposition 1 is necessary for a Pareto improvement, it does not suffice to consider pure transfers between different generations [as in Chattopadhyay and Gottardi [9]] that leave the capital stock unaffected. An improving allocation may e.g. involve an increase in the capital stock combined with transfers from young to old at certain date-events. 4 Another possibility is that improvements start with a decrease in capital. 5 So under uncertainty, there are many possible changes in consumption and investment and it is tedious to rule them out separately. To circumvent the problem of ruling out every possible change in the consumption and investment plan separately, our proof focuses on deriving an expression formally equivalent to the non-vanishing Gaussian curvature condition for pure consumption changes used by Chattopadhyay and Gottardi [9] that holds for all possible joint deviations. This is accomplished by starting 4

Under certainty, such increases in the capital stock can be ruled out directly as

an additional possibility for improvements from first order conditions of households and firms and convexity of preferences and technology. This is not the case under uncertainty. Also, under certainty, by a similar argument, increases in young age consumption can be easily ruled out. 5 This possibility arises as additional source of improvements in the certainty and uncertainty case.

11

with the non-vanishing Gaussian curvature condition for pure consumption changes and incorporating a quadratic estimate from a second order Taylor approximation for the production changes as used in Zilcha [23] into this equation. 6 The first order conditions of the firm relating state prices and the marginal product of capital eliminates the linear capital term. The quadratic terms in consumption and capital can be simplified to a single quadratic expression in joint consumption-investment changes. 7 The resulting expression is formally equivalent to the one in Chattopadhyay and Gottardi [9] and their generalized Cass type argument can be applied. 8 It results that the characterization of interim Pareto optimality with production is the same as in the pure exchange case.

4

Testability: Observable Implications of Optimality

The Rate of Return to Capital and Dynamic Efficiency We now examine how the accumulation of capital interacts with opportunities of intergenerational risk sharing in a dynamic economy. Dynamic efficiency rules out the overaccumulation of capital in the sense that a decrease in savings would allow for a permanently higher aggregate consumption level [see Dechert and Yamamoto [10] and Zilcha [23]. Zilcha [23] shows in his Lemma 1, p.373, that this definition is equivalent to his original notion of dynamic efficiency given on p. 369 of his paper.]

6 7

See equations (A.1) and (A.2) in the proof. This is achieved by using an elementary inequality, see the paragraph preceding

equation (A.3). 8 See equation (A.5) in the proof.

12





Definition 5 A sequence of investment decisions k (σ)σ∈Γ is dynamically 

efficient if there exists no other sequence of investment decisions kb (σ)σ∈Γ



such that (with at least one strict inequality): 



∀σ ∈ Γ: f kb (σ) , θ − kb (σ 0 ) ≥ f (k (σ) , θ) − k (σ 0 )

∀σ 0 = (σ, θ) ∈ σ +

f (k−1 , θ0 ) − kb (σ0 ) ≥ f (k−1 , θ0 ) − k (σ0 )

Let us introduce the risky rate of return r∗ , which is defined as the marginal 

 







∞ ∞ , θ = σs∞ . , θ , where σs−1 product of capital, r∗ (σs∞ ) = f 0 k σs−1

For a given subtree Γ (σe ) (that contains all successor nodes of σe ), we define the set of paths in Γ (σe ) as the set of paths σ ∞ with σt∞eσ = σe . The following ( ) characterization of dynamic efficiency under uncertainty is - apart from a technicality concerning the uniform bound - due to Zilcha [23]: Proposition 2 A feasible allocation is dynamically inefficient if and only if there exists a node σe ∈ Γ and some C > 0 such that along every path σ ∞ in Γ (σe ) we have t ∞ Y X

r∗ (σs∞ ) ≤ C.

(4)

t=1 s=1

By definition, interim Pareto optimality implies dynamic efficiency. In fact, in a deterministic framework the two notions of efficiency coincide [see also Bose and Ray [6]]: under certainty the condition for interim Pareto suboptimality in Proposition 1 reduces to

∞ Q P t t=1

s=1 rs

< ∞. This condition is equiv-

alent to the deterministic Cass criterion for dynamic inefficiency. However, under uncertainty dynamic efficiency is a strictly weaker efficiency benchmark [see Bertocchi [3,4] for a discussion]. Comparing conditions (3) and (4), the condition for dynamic inefficiency is a special case of the one for interim 13

Pareto suboptimality for the choice of a transfer pattern starting in σe given by λ (σ 0 ) :=

p(σ 0 ) p(σ)

· f 0 (k (σ) , θ) for σ 0 = (σ, θ) ∈ Γ (σe ) . Curing dynamic inefficiency

focuses attention on a particular form of improvement: transfers with payoffs f 0 in each state. Arbitrary Returns and Pareto Optimality More generally, consider an arbitrary (portfolio of) asset(s) with given return R (σ) , σ ∈ Γ. In a competitive equilibrium, by no-arbitrage, this return has to satisfy the Euler equation: X p (σ 0 ) σ 0 ∈σ +

p (σ)

· R (σ 0 ) = 1 for each σ ∈ Γ.

(5)

We will call each function R : Γ → R such that (5) holds a return. Since markets are sequentially complete, each return corresponds to the rate of return of an asset that could be generated from a given set of assets that span the market space. A particular return is given by the marginal product of capital f 0 , the risky rate of return, which satisfies (5) in an interior competitive equilibrium from the profit maximization condition of the firm. Endow the shock space S with the discrete topology and endow the space Q∞

t=0

St with St = S for each t ≥ 1, with the corresponding product topology.

Since we can identify each path σ ∞ with a sequence of shocks (θ0 , θ1 , θ2 , ...) , this defines a topology on the set of all paths of the date-event tree Γ. For a given transfer pattern λ, let Γλ := {σ ∈ Γ |λ (σ) > 0}. Note that our definition of a transfer pattern implies that Γλ is a subtree. We identify Γλ with the paths it contains. We have for a general class of returns:

Proposition 3 Consider an interior competitive equilibrium allocation which is interim suboptimal. Then for any transfer pattern λ that satisfies the suboptimality condition (3) and any return R (σ) ≥ 0, σ ∈ Γ, such that for each 14

σ ∈ Γλ there is some σ 0 ∈ σ + ∩ Γλ with R (σ 0 ) > 0, we have: (a) The set of paths σ ∞ in Γλ with

P∞ Qt t=1

s=1

R (σs∞ ) < ∞ is nonempty.

(b) If Γλ contains an infinite number of paths, the set of paths σ ∞ in Γλ with P∞ Qt t=1

s=1

R (σs∞ ) < ∞ is not finite.

(c) For every node σt ∈ Γλ we have σt = σt∞ for some σ ∞ with P∞ Qt t=1

s=1

R (σs∞ ) < ∞.

(d) The set paths σ ∞ with

P∞ Qt t=1

s=1

R (σs∞ ) < ∞ is dense in Γλ (with respect

to the relative topology induced by the product topology on the set of all paths Γ).

(c) states that paths with

P∞ Qt t=1

s=1

R (σs∞ ) < ∞ are spread out ”uniformly”

across the subtree on which the improving transfers exist. (d) restates (c) in the language of point set topology. The following proposition yields tests for optimality and suboptimality. Proposition 4 (a) If at an interior competitive equilibrium allocation there is a nonnegative return R (i.e. R (σ) ≥ 0 for all σ ∈ Γ) such that for all paths σ ∞ in Σ we have

P∞ Qt t=1

s=1

R (σs∞ ) = ∞, then it is interim Pareto

optimal. (b) If at an interior competitive equilibrium allocation for some return R and some node σe ∈ Γ we have that along every path σ ∞ in Γ (σe ) , P∞

t=t(e σ )+1

Qt

s=t(e σ )+1

R+ (σs∞ ) ≤ M for some M > 0, then it is interim

Pareto suboptimal. Here, R+ (σ) := max {R (σ) , 0} .

Proposition 4 (a) is just the logical contraposition to Proposition 3 (a). Another somewhat surprising implication of Proposition 4 (a) is that that all competitive equilibrium allocations whose state prices are consistent with the 15

return (i.e. for which (5) holds), are interim Pareto optimal once we have found a return R for which

P∞ Qt t=1

s=1

R (σs∞ ) = ∞ for all paths σ ∞ .

First, we consider general assets with nonnegative returns while Chattopadhyay [8] considers only a restricted class of long-lived assets. Second, our test may yield an optimality conclusion when Chattopadhyay’s test is inconclusive, but not vice versa. 9 In comparison to our Proposition 4 (b), Theorem 2 of Chattopadhyay [8] assumes an additional bound for

Qt

s=t(e σ )+1

R (σs∞ ) for the case of negative

returns (see condition (b) of Theorem 2 in Chattopadhyay [8]). This assumption is, however, not needed to conclude for suboptimality under sequentially complete markets which is ours and Chattopadhyay’s [8] market structure. It is possible to construct examples where

P∞ Qt t=1

 





0 ∞ k σs−1 , θ diverges s=1 f

almost surely, 10 but where the competitive equilibrium is nevertheless Pareto suboptimal: 11 P∞ ∗ ∞ Chattopadhyay considers whether t=1 r t diverges, where r t ≤ q (σ0 ) · Qt ∞ inf s=1 R (σs ) , where R is the return of the corresponding (portt ∞ ∞ (σ1 ,...,σt )∈×s=1 Σs P∞ folio of) long-lived asset(s) and q ∗ (σ0∞ ) its period zero price. Clearly, if t=1 r t P Qt ∞ diverges, we also have ∞ t=1 s=1 R (σs ) = ∞ for every path. But the converse 9

need not hold, i.e. our test may indicate Pareto optimality while his test remains inconclusive. 10 Almost surely refers to the unique probability measure on the set of paths Σ that, by the Kolmogorov extension theorem, is generated from the transition probabilities qt+1 (θt+1 |σ ) for all t ≥ 0 and the ”initial” probability q (σ0 ) = 1. 11 See the discussion paper version of the paper or a technical appendix available upon request. Proposition 3 (d) states that in a Pareto suboptimal competitive equilibrium allocation the set of paths where this sum converges is dense. Both

16

Remark 2 For any given asset return, the divergence requirement in Proposition 4 (a) cannot be weakened to require divergence of the sum ”almost surely” instead of ”on every path” if one wants to ensure interim Pareto optimality of a competitive equilibrium.

Appendix

A

Proofs

Preliminary Results

We will make use of the so-called Gaussian curvature conditions on utility functions. In our set-up, these conditions can be shown to hold under our primitive assumptions on utility functions and the interiority assumptions on competitive equilibria. A proof is available upon request. We only state the non-vanishing Gaussian curvature condition. The bounded Gaussian curvature condition is omitted. It is needed to show that the condition stated in the characterization implies the existence of Pareto improvements. Preferences are given by Uσ (c (σ)) = Eσ u (c (σ)) =

P

θt+1 ∈S

qt+1 (θt+1 | σ) · u (cy (σ) , co ((σ, θt+1 ))) .

• non-vanishing Gaussian curvature: there exists a ρ> 0 and a δ > 0, such that for any interior competitive equilibrium (c∗ , k ∗ , p) , for all feasible 



allocations cb, kb ∈ K 0 , where K 0 is a convex, compact set (in the product statements are consistent. Together they just say that denseness of the set of convergent paths does not even imply convergence on a set of positive measure.

17

topology defined on

Q

σ∈Γ

R3+ ) that contains the set of competitive alloca-

tions, ∀σ ∈ Γ, Uσ (cb (σ)) ≥ Uσ (c (σ)) =⇒

(δ1 (σ))2 δ2 (σ , σ) ≥ −δ1 (σ) + ρ p (σ) σ 0 ∈σ + 0

X

where δ1 (σ) = p (σ)·[cby (σ) − c∗y (σ)] and δ2 (σ 0 , σ) = p (σ 0 ) · [cbo (σ 0 ) − c∗o (σ 0 )] for σ 0 ∈ σ + . ρ is called the lower curvature coefficient.

Proposition 1

Proof. If condition (3) in the proposition holds it is straightforward from Chattopadhyay and Gottardi [9], Theorem 2, that the allocation is not interim Pareto optimal. It therefore remains to be shown that the existence of an allocation that improves upon a competitive allocation implies the condition in the characterization. Our proof highlights and isolates the new steps in comparison to the pure exchange setup. b improving upon the competitive allocation (c∗ , k ∗ ) Suppose an allocation (cb, k)

exists. Since the economy is bounded above and strictly convex, it can be b lies in the comshown that we can assume that the deviating allocation (cb, k)

pact set K 0 defined above. That this is possible follows from the way K 0 is constructed in the derivation of the non-vanishing Gaussian curvature condition. This construction is straightforward and available upon request. By the non-vanishing Gaussian curvature condition we have, for any node σ: X

p (σ 0 )·[cbo (σ 0 ) − c∗o (σ 0 )] ≥ −p (σ)·[cby (σ) − c∗y (σ)]+ρ·p (σ)·(cby (σ) − c∗y (σ))2 .

σ 0 ∈σ +

(A.1) 



Define ∆f (σ 0 ) := f kb (σ) , θ − f (k ∗ (σ) , θ) , ∆k (σ) := kb (σ) − k ∗ (σ) and

18

∆cy (σ) := cby (σ) − c∗y (σ). Further, define ∆t (σ) := − (cby (σ) − c∗y (σ)) − 



kb (σ) − k ∗ (σ) . Thus ∆t (σ) is the combined change in young age consump-

tion and in saving. ∆te(σ 0 ) := cbo (σ 0 ) − c∗o (σ 0 ) − ∆f (σ 0 ) denotes the change in old age consumption and production at node σ 0 . Using this we get from (A.1): X





p (σ 0 )· ∆f (σ 0 ) + ∆te(σ 0 ) ≥ p (σ)·(∆k (σ) + ∆t (σ))+ρ·p (σ)·(∆cy (σ))2 .

σ 0 ∈σ +





Note that by a Taylor expansion f kb (σ) , θ − f (k ∗ (σ) , θ) = f 0 (k ∗ (σ) , θ) · h

i

2



kb (σ) − k ∗ (σ) + 21 ·f 00 (ζ, θ)· kb (σ) − k ∗ (σ)





for some ζ(σ) ∈ kb (σ) , k ∗ (σ) .

By assumption, kb (σ) and k ∗ (σ) are from a compact set (being the coordinatewise projection of the compact set K 0 ) and therefore there exists some constant c > 0 such that 1 f kb (σ) , θ −f (k ∗ (σ) , θ) ≤ f 0 (k ∗ (σ) , θ)· kb (σ) − k ∗ (σ) − ·c· kb (σ) − k ∗ (σ) 2 



h

i



2

.

Thus X

p (σ 0 ) · ∆te(σ 0 ) +

σ 0 ∈σ +

p (σ 0 ) · f 0 (k ∗ (σ) , θ) · ∆k (σ)

X

(A.2)

σ 0 ∈σ +

1 X ≥ p (σ) · (∆k (σ) + ∆t (σ)) + ρ · p (σ) · (∆cy (σ))2 + · p (σ 0 ) · c · (∆k (σ))2 . 2 σ0 ∈σ+ Using the first order condition from profit maximization

P

p (σ 0 )·f 0 (k ∗ (σ) , θ) =

σ 0 ∈σ +

p (σ) , and that f 0 (k ∗ (σ) , θ) ≤ ν by the interiority of the allocation for some ν > 0, we obtain: 1 p (σ) p (σ 0 )·∆te(σ 0 ) ≥ p (σ)·(∆t (σ))+ρ·p (σ)·(∆cy (σ))2 + ·c· (∆k (σ))2 . 2 ν σ 0 ∈σ + X

Note that (∆cy (σ))2 + (∆k (σ))2 ≥ 1 2

n

1 2

· (∆cy (σ) + ∆k (σ))2 . Defining γ :=

o

· min ρ, 2νc , we thus have X

p (σ 0 ) · ∆te(σ 0 ) ≥ p (σ) · (∆t (σ)) + γ · p (σ) · (∆t (σ))2

σ 0 ∈σ +

19

(A.3)

Note that from the resource constraint: −∆cy (σ 0 ) − ∆k (σ 0 ) = ∆co (σ 0 ) − ∆f (k (σ) , θ) ,

(A.4)

where ∆co (σ 0 ) = cbo (σ 0 ) − c∗o (σ 0 ) . Thus ∆t (σ) = ∆te(σ) for each node σ, and we can use the new quadratic estimate (A.3) to obtain: X

p (σ 0 ) · ∆t (σ 0 ) ≥ p (σ) · (∆t (σ)) + γ · p (σ) · (∆t (σ))2 .

(A.5)

σ 0 ∈σ +





Since the allocation cb, kb Pareto improves upon (c∗ , k ∗ ) , there must exist some node σ ∗ with ∆t (σ ∗ ) > 0. From here on one can follow the arguments in the proof of Theorem 1, pp. 53 in Chattopadhyay and Gottardi [9] to complete the proof.

Proposition 3

Proof. If we define for σ ∈ Γλ and for σ 0 = (σ, θ) ∈ Γλ : p(σ 0 ) p(σ)

· R (σ 0 ) µ (σ ) := P p(e σ) e) e σ ∈Γλ ∩σ + p(σ) · R (σ 0

we have that µ is well-defined since by assumption R (σe ) > 0 for some σe ∈ Γλ ∩ σ + . Further

P

σ 0 ∈Γλ ∩σ +

µ (σ 0 ) = 1 for all σ ∈ Γλ . Since the µ (σ 0 ) are also

nonnegative, they form a transfer pattern according to Definition 4. For any transfer pattern λ and for each node σ ∈ Γλ there must exist an σ 0 ∈ Γλ ∩ σ + such that λ (σ 0 ) ≥ µ (σ 0 ) . (If not,

P

σ 0 ∈Γλ ∩σ +

λ (σ 0 ) < 1, which contradicts

the definition of a transfer pattern). Starting from the root σ0 , we can thus inductively define a path σ ∞ in Γλ such that λ (σt∞ ) ≥ µ (σt∞ ) for all t. Now if λ is an improving transfer pattern, then on this path we have for some A > 0 A≥

∞ Y t X t=1 s=1

λ (σs∞ ) ·

∞ Y t X 1 1 ≥ µ (σs∞ ) · ∞ p (σt ) t=1 s=1 p (σt∞ )

20

=

t=1 s=1

=

p(σs∞ ) ∞ p(σs−1 )

∞ Y t X

· R (σs∞ )

P p(e σ) + · R (σe ) ∞ ∞ e σ ∈Γλ ∩(σs−1 ) p(σs−1 )

∞ Y t X

1

R (σs∞ ) · P t=1 s=1

+

∞ e σ ∈Γλ ∩(σs−1 )

p(e σ)

∞ p(σs−1 )

·

1 p (σt∞ )

≥ · R (σe )

∞ Y t X

R (σs∞ )

t=1 s=1

where the second equality uses the normalization p (σ0 ) = 1 and the last inequality uses the fact that

P p(e σ) + · R (σe ) ≤ 1 from (5) and ∞ ∞ e σ ∈Γλ ∩(σs−1 ) p(σs−1 )

the nonnegativity of R. This proves the first claim. Since for each σt ∈ Γλ , the subtree Γλ ∩ Γ (σt ) also allows for an improving transfer pattern, the third claim follows from repeating the arguments above. The fourth claim follows immediately from the third claim since the product topology is characterized by pointwise convergence. Since the number of nodes in Γλ with more than one successor node is at least countable if the number of paths in Γλ is not finite, the second claim follows.

Proposition 4

Proof. (a) If the condition

P∞ Qt t=1

R (σs∞ ) = ∞ holds for all paths σ ∞ , we

s=1

have R (σ) > 0 for all σ ∈ Γ and thus the assumptions of Proposition 3 hold. If an improvement is possible, by Proposition 3, part (a), there exists at least one path σ ∞ on which

P∞ Qt t=1

s=1

R (σs∞ ) < ∞. This yields a contradiction.

(b) Since R is a return, we have for each σ ∈ Γ that Hence we have for each σ ∈ Γ that p(σ 0 ) p(σ)

p(σ 0 ) σ 0 ∈σ + p(σ)

P

p(σ 0 ) σ 0 ∈σ + p(σ)

P

· R (σ 0 ) = 1.

· R+ (σ 0 ) ≥ 1. Define µe (σ 0 ) =

0 · R+ (σ 0 ) for each σ 0 ∈ σ + with σ ∈ Γ (σe ) and µ (σ 0 ) = P µe(σ )µe bσ . For ( ) b σ ∈σ +

each σ ∈ σe − ∪ {σe } define µ (σ) = 1 and µ (σ) = 0 for all remaining σ ∈ Γ. We have 0 ≤ µ (σ 0 ) ≤ µe (σ 0 ) and

P

σ 0 ∈σ +

µ (σ 0 ) = 1 for each σ 0 ∈ σ + with

σ ∈ Γ (σe ) . Now for each path σ ∞ with σt∞eσ = σe , ( ) 21

1 

p σt∞eσ ( )

∞ X



t Y

∞ X

R+ (σs∞ ) =

t=t(e σ )+1 s=t(e σ )+1

t Y

t=t(e σ )+1 s=t(e σ )+1 ∞ X

=

t=t(e σ )+1 ∞ X



t=t(e σ )+1

Because

P∞

t=t(e σ )+1

we have that  p

Qt

1 σ∞

s=t(e σ )+1

 ·

p (σs∞ ) 1 · R+ (σs∞ ) · ∞ p (σs−1 ) p (σt∞ )

1 p (σt∞ ) 1 p (σt∞ )

t Y

µe (σs∞ )

s=t(e σ )+1 t Y

µ (σs∞ )

s=t(e σ )+1

R+ (σs∞ ) ≤ M for each path σ ∞ with σt∞eσ = σe , ( )

P∞

t=t(e σ )+1

Qt

s=t(e σ )+1

R+ (σs∞ ) ≤  p

M

 for all those

σ∞

t(e σ) (e) paths and thus, from the inequality proved above and the definition of the t σ

transfer P∞

1 t=1 p(σ ∞ ) t

pattern Qt

s=1

µ (σs∞ ) ≤  p

M

 +

µ, σ) Pt(e

we

1 t=1 p(σ ∞ ) t

σ∞

have

for all paths σ ∞ ∈ Σ. It

(e)

t σ

follows from Proposition 1 for B =  p

M σ∞

+

σ) Pt(e

1 t=1 p(σ ∞ ) t

that the competi-

(e) tive equilibrium is interim Pareto suboptimal. t σ

References

[1] Allais, Maurice (1947), ”Economie et Int´erˆet”, Imprimerie National, Paris. [2] Balasko, Yves and Karl Shell (1980), ”The Overlapping-Generations Model, I: The Case of Pure Exchange without Money”, Journal of Economic Theory, 23, 281-306. [3] Bertocchi, Graziella (1991), ”Bubbles and Inefficiencies”, Economics Letters, 35, 117-122. [4] Bertocchi, Graziella (1994), ”Safe Debt, Risky Capital”, Economica 61, 493-508. [5] Blanchard, Olivier-Jean and Philippe Weil (2001), ”Dynamic Efficiency, The Riskless Rate, and Debt Ponzi Games under Uncertainty”, MIT, Department

22

of Economics Working Paper 01-41. [6] Bose, Amitava and Debraj Ray (1993), ”Monetary equilibrium in an overlapping generations model with productive capital”, Economic Theory, 3, 697-716. [7] Cass, David (1972), ”On capital overaccumulation in the aggregative, neoclassical model of economic growth: A complete characterization”, Journal of Economic Theory, 4, 200-223. [8] Chattopadhyay, Subir (2006), ”Optimality in stochastic OLG models: Theory for Tests”, Journal of Economic Theory, 131, 282-294. [9] Chattopadhyay, Subir and Piero Gottardi (1999), ”Stochastic OLG Models, Market Structure, and Optimality”, Journal of Economic Theory, 89, 21-67. [10] Dechert, W. Davis and Kenji Yamamoto (1992), ”Asset Valuation and Production Efficiency in an Overlapping-Generations Model with Production Shocks”, Review of Economic Studies, 59, 389-405. [11] Demange, Gabrielle and Guy Laroque (1999), ”Social Security and Demographic Shocks”, Econometrica, 67, 527-42. [12] Demange, Gabrielle and Guy Laroque (2000), ”Social Security, Optimality, and Equilibria in a Stochastic Overlapping Generations Economy”, Journal of Public Economic Theory, 2, 1-23. [13] Diamond, Peter (1965), ”National Debt in a Neoclassical Growth Model”, American Economic Review, 55, 1127-1155. [14] Gottardi, Piero and Felix Kubler (2006), ”Social Security and Risk Sharing”, Working Paper. [15] Manuelli, Rodolfo (1990), ”Existence and Optimality of Currency Equilibrium in Stochastic Overlapping Generations Models: The Pure Endowment Case”, Journal of Economic Theory, 51, 268-94.

23

[16] Muench, Thomas (1977), ”Optimality, the Interaction of Spot and Futures Markets, and the Nonneutrality of Money in the Lucas Model”, Journal of Economic Theory, 15, 325-344. [17] Malinvaud, Edmond (1953), ”Capital Accumulation and Efficient Allocation of Resources”, Econometrica, 21, 233-273. [18] Okuno, Masahiro and Itzhak Zilcha (1980), ”On the Efficiency of a Competitive Equilibrium in Infinite Horizon Monetary Economies”, Review of Economic Studies, 47, 797-807. [19] Peled, Dan (1982), ”Informational Diversity over Time and the Optimality of Monetary Equilibria”, Journal of Economic Theory, 28, 255-74. [20] Phelps, Edmund S. (1961), ”The Golden Rule of Accumulation: A Fable for Growthmen”, American Economic Review, 55, 638-643. [21] Samuelson, Paul A. (1958), ”An exact consumption-loan model of interest with or without the contrivance of money”, Journal of Political Economy, 66, 467482. [22] Tirole,

Jean

(1985),

”Asset

Bubbles

and

Overlapping

Generations”,

Econometrica, 53, 1499-1528. [23] Zilcha, Itzhak (1990), ”Dynamic Efficiency in Overlapping Generations Models with Stochastic Production”, Journal of Economic Theory, 52, 364-79.

24

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