TCER WORKING PAPER SERIES GOLDEN RULE OPTIMALITY IN STOCHASTIC OLG ECONOMIES. Eisei Ohtaki. June 2012

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1 TCER WORKING PAPER SERIES GOLDEN RULE OPTIMALITY IN STOCHASTIC OLG ECONOMIES Eiei Ohtaki June 2012 Working Paper E-44 TOKYO CENTER FOR ECONOMIC RESEARCH Iidabahi, Chiyoda Tokyo, Japan c 2012 by Eiei Ohtaki. All right reerved. Short ection of text, not to exceed two paragraph, may be quoted without explicit permiion provided that full credit, including c notice, i given to the ource.

2 Abtract Thi tudy examine conditional golden rule optimality (CGRO) in tochatic overlapping generation model to complement the exiting reult on conditional Pareto optimality (CPO). Although an example in which CPO implie CGRO i preented, it i hown that uch a ituation i avoidable under trictly convex preference. Under uch preference, both CPO and CGRO are characterized by the condition on the dominant root for the agent common matrix of marginal rate of ubtitution. We demontrate that CGRO require the dominant root being exactly equal to one, wherea CPO allow it to be le than one. By adopting CGRO rather than CPO, we provide welfare theorem in the financial economy. Eiei Ohtaki TCER and Faculty of Economic Keio Univerity Mita , Minato-ku, Tokyo , Japan ohtaki@g.econ.keio.ac.jp

3 Golden Rule Optimality in Stochatic OLG Economie Eiei Ohtaki Faculty of Economic, Keio univerity, Mita , Minato-ku, Tokyo , Japan addre: Firt Draft: Augut 30, 2011 Current Draft: June 1, 2012 Abtract: Thi tudy examine conditional golden rule optimality (CGRO) in tochatic overlapping generation model to complement the exiting reult on conditional Pareto optimality (CPO). Although an example in which CPO implie CGRO i preented, it i hown that uch a ituation i avoidable under trictly convex preference. Under uch preference, both CPO and CGRO are characterized by the condition on the dominant root for the agent common matrix of marginal rate of ubtitution. We demontrate that CGRO require the dominant root being exactly equal to one, wherea CPO allow it to be le than one. By adopting CGRO rather than CPO, we provide welfare theorem in the financial economy. Keyword: Golden rule optimality; Pareto optimality; Dominant root; Welfare theorem; Stochatic overlapping generation model. JEL Claification Number: D50; D62; D81; E40.

4 1 Introduction In the overlapping generation (OLG) economy, competitive equilibrium might not achieve an optimal allocation, even when market operate perfectly, a in the Arrow-Debreu abtraction. It i undertood that thi ort of inefficiency i caued by the lack of a tranverality condition at infinity. To deign active policie (a ocial ecurity) which remedy thi type of inefficiency, it i important to identify optimality with eaily verifiable condition. Many of tudie have characterized optimality in tochatic OLG model: Peled (1984), Aiyagari and Peled (1991), Manuelli (1990), Chattopadhyay and Gottardi (1999), Chattopadhyay (2001, 2006), and Bloie and Calciano (2008) are uch example. 1 However, determinitic model have two primary optimality criteria: Pareto optimality and golden rule optimality. The above-mentioned tudie focued on conditional Pareto optimality (CPO) and ignored conditional golden rule optimality (CGRO). Therefore, to complement the exiting reult on CPO, it eem worthwhile to tudy CGRO. To do o, thi tudy conider a imple, but rather, canonical tochatic OLG model uch a that tudied by Aiyagari and Peled (1991) and Chattopadhyay (2001). We then introduce two optimality criteria, CPO and CGRO for tationary feaible allocation. 2 According to thee criteria, agent welfare i evaluated by conditioning their utility on the tate at the date of their birth. Agent are thu ditinguihed not only by their type and date of birth but alo by the tate at that date, and an agent preference i defined over a et of contingent conumption tream available in the two period of that agent lifetime. The difference between thee two criteria i clear: CPO cope with the welfare of the initial old, while CGRO doe not. Thi tudy firt dicue the relationhip between CPO and CGRO. It i often believed that CGRO implie CPO and the et of CGRO allocation i trictly maller than that of CPO allocation. However, thi tudy preent two example violating thee intuition. One of two example i a ituation wherein CGRO doe not imply CPO. The other i a ituation wherein the et of CPO and CGRO allocation coincide with each other, and thu there i no CPO allocation which i not CGRO. We demontrate that thee two anomalou ituation are avoidable by impoing, for example, trict quai-concavity and boundary condition on lifetime utility function. 1 Thee tudie conidered pure-endowment model, wherea Demange and Laroque (1999, 2000), Barbie et al. (2007), and Gottardi and Kubler (2011) tudied model with production. 2 CPO wa firt propoed by Muench (1977). 1

5 Under trict quai-concavity of lifetime utility function, thi tudy characterize both CPO and CGRO of tationary feaible allocation, wherea the exiting literature focue on the CPO of (tationary) equilibrium allocation. A an analogue of Pareto optimality in the tandard Arrow-Debreu economy, the all agent matrixe of marginal rate of ubtitution mut coincide with one another at a CPO or CGRO allocation. Although CPO of a tationary feaible allocation i characterized by the dominant root of the matrix aociated to the allocation being le than or equal to unity, we find that CGRO of a tationary feaible allocation i characterized by the dominant root of the matrix being exactly equal to unity. By thee characterization, we might ay that CGRO i tronger than CPO a an optimality criterion. It i known that a tationary equilibrium allocation with valued money, if any, i CPO. 3 By applying our reult to the iue of equilibrium welfare, thi tudy conclude that a tationary equilibrium allocation with valued money, if any, i not only CPO but alo CGRO. Thi can be interpreted a the firt welfare theorem of the tochatic OLG economy with money. By the firt welfare theorem, we note that there might exit a CPO allocation that cannot be implemented by any tationary monetary equilibrium with tranfer. That i, the econd welfare theorem might not hold when we adopt CPO a an optimality criterion. By adopting CGRO, not CPO, thi tudy alo provide the econd welfare theorem, i.e., any interior CGRO allocation can be achieved by a tationary monetary equilibrium under certain lump-um tranfer. The organization of thi paper i a follow: Section 2 preent detail of the model. Section 3 define CPO and CGRO and dicue the relationhip between thee two optimality criteria. Section 4 characterize CPO and CGRO for tationary feaible allocation under trict quaiconcavity of utility function. Section 5 introduce tationary equilibrium to the model and applie reult given in the previou ection to equilibrium allocation. Section 6 preent welfare theorem in the economy with financial aet. Section 7 conclude the paper. Proof of reult are provided in the Appendix. 2 The Economy Thi tudy conider a tationary, one-good, finite-tate, pure-endowment tochatic overlapping generation model with finitely lived agent, a tudied by Aiyagari and Peled (1991) and Chattopadhyay (2001). Time i dicrete and run from t = 0 to infinity. Uncertainty i 3 See alo Sakai (1988). 2

6 modeled by a tationary Markov proce with it finite tate pace S uch that 0 S. For each t 0, we denote by t the tate realized in period t, called period t tate, where the initial tate 0 S i treated a given. 4 After the realization of the tate in each period t 1, a new generation, the member of which are called newly born agent or imply agent, i born and live for two period. Let H be a nonempty finite et of member of each generation. We will aume that the economy i tationary, i.e., the endowment and preference tructure of each agent depend only on the realization of the Markov tate during hi/her lifetime, not on time or on pat realization. Thu, (i) the endowment tream of each agent h H born at tate S i denoted by ω h = (ω h1, (ω h2 ) S) R + R S + and (ii) hi/her lifetime utility function i denoted by U h : R + R S + R, where ω h1 and (ω h2 ) S decribe the endowment at birth and all tate in the following period. It i aumed that ω h 0 and U h i trictly monotone increaing, quai-concave, and continuouly differentiable on the interior of it domain. 5 In addition, a one-period lived generation, the member of which are called initial old agent or imply initial old, i born after the realization of period 1 tate 1 S. The et of the initial old i given by H a defined above. Each initial old h H born at period 1 tate 1 i aumed to be endowed with ω 2h 0 1 := ω 2h 0 1 unit of the conumption good in hi/her lifetime and hi/her conumption tream c h2 0 1 R + i ranked according to a utility function u 0 (c h2 0 1 ) := c h Let S 0 := {0} S and ω := h H (ω1h + ω2h ) for each (, ) S 0 S, which i the total endowment when the current and preceding tate are and, repectively. 6 We concentrate our attention not on all feaible allocation but on tationary feaible allocation. A tationary feaible allocation of thi economy i a family c = {c h1, c h2 } h H of function c h1 : S R + and c h2 : S 0 S R + uch that ( (, ) S 0 S) h H c h1 + h H c h2 = ω, where c h2 0 1 R + i the conumption of the initial old h born at period 1 tate 1, and c h = (c h1, (c h2 ) S) R + R S + i the conumption tream of the agent h born at the Markov tate. Let A be the et of all tationary feaible allocation with it generic element c. Note that A i 4 Thi tudy implicitly conider a tandard date-event tree a een in, for example, Chattopadhyay (2001). Therefore, the initial tate 0 can be interpreted a the root of the date-event tree. 5 In the ret of thi tudy, we denote by U1 h (c 1, c 2 ) and U h (c1, c 2 ) the partial derivative U h (c 1, c 2 )/ c 1 and U h (c 1, c 2 )/ c 2 for all h H, all, S, and all (c 1, c 2 ) R + R S +, repectively. 6 We introduce S 0 to tell the conumption of the initial old from the agent conumption in the econd period of her life. 3

7 nonempty, bounded, cloed, and convex. A tationary feaible allocation c i interior if c h1 > 0 and c h2 > 0 for all, S and all h H. 3 Optimality Criteria Thi ection introduce two criteria of optimality of tationary feaible allocation, CPO and CGRO, and we dicu the relationhip between thee two criteria. For any two tationary feaible allocation b, c A, we ay that b CPO-dominate c if ( (h, ) H S) b h2 0 ch2 0, U h (b h ) U h (c h ) with trict inequality omewhere. CPO i then defined a follow: Definition 1 A tationary feaible allocation c i aid to be conditionally Pareto optimal if there exit no other tationary feaible allocation b that CPO-dominate c. CPO conider the welfare of the initial old, wherea CGRO ignore it. For any two tationary feaible allocation b, c A, we ay that b CGRO-dominate c if ( (h, ) H S) U h (b h ) U h (c h ) with trict inequality omewhere. CGRO i then defined a follow: Definition 2 A tationary feaible allocation c i aid to be conditionally golden rule optimal if there exit no other tationary feaible allocation b that CGRO-dominate c. Thee two criteria are analogue to thoe in a determinitic environment. In the ret of thi tudy, we often call a tationary CPO (CGRO) allocation, a CPO (CGRO) allocation. Let CP O and CGRO be the et of CPO and CGRO allocation, repectively. Next we dicu the relationhip between CPO and CGRO. It i often conidered that CP O CGRO, i.e., a CGRO allocation i alo CPO. However, the following example demontrate that CGRO doe not necearily imply CPO. Example 1 (CP O CGRO ) The firt example illutrate an anomalou ituation uch that a CGRO allocation i not necearily CPO. Conider the economy uch that S = {α, β} and H i a ingleton. In thi example, we ignore upercript h (of endowment, conumption, and preference), becaue H i a ingleton. Suppoe that ω 2 i independent of the current tate for 4

8 each, S. Note that, in thi economy, we can rewrite the total endowment a ω ω for all, S. Alo let U (c ) = 2c 1 +c 2 α+c 2 β for each S. In the current etting, one can eaily verify that, for every tationary feaible allocation c, it hold that U α (c ) = ω α + ω β +c 1 α c 1 β and U β (c ) = ω α + ω β c 1 α + c 1 β. Define the tationary feaible allocation x = {x1, x 2 } by x 1 = ω and x 2 = 0 for each (, ) S 0 S. Obviouly, allocation x i well-defined and CGRO, not CPO. In fact, for ufficiently mall ε > 0, define the tationary feaible allocation y = {y 1, y 2 } by y 1 = ω ε and y2 = ε for each (, ) S 0 S. Thi allocation i well-defined. Further, y CPO-dominate x, becaue it improve the initial old welfare. Thi example, urpriingly, alo illutrate a ituation uch that CP O CGRO and CP O CGRO, i.e., the et of CPO allocation i a proper ubet of the et of CGRO allocation. In fact, all tationary feaible allocation are CGRO and the et of CPO allocation i given by {c A : c 1 α = 0} {c A : c 1 β = 0} in thi example. The following propoition provide a ufficient condition for avoiding ituation illutrated in Example 1. Propoition 1 CP O CGRO, i.e., every conditionally golden rule optimal allocation i conditionally Pareto optimal if either (a) for any x = (x 1, x 2 ), y = (y 1, y 2 ) R + R S + with x 1 y 1, any α (0, 1), and any (h, ) H S, U h (αx + (1 α)y) > min{u h (x), U h (y)}; or (b) for any x = (x 1, x 2 ), y = (y 1, y 2 ) R + R S + with x 2 y 2, any α (0, 1), and any (h, ) H S, U h (αx + (1 α)y) > min{u h (x), U h (y)} hold. Proof of Propoition 1. See the Appendix. Q.E.D. Note that both condition (a) and (b) provided in thi propoition hold, for example, under trict quai-concavity of lifetime utility function. Thu, a a corollary of Propoition 1, we can oberve that CP O CGRO, provided that lifetime utility function are trictly quaiconcave. We can now ay that CGRO i not a weaker criterion of optimality than CPO under trict quai-concavity of lifetime utility function. We have provided ufficient condition that enure that CGRO implie CPO. The natural quetion following thi reult i whether CPO implie CGRO under uch condition. Although 5

9 it may eem that there alway exit a CPO allocation that i not CGRO, the following example illutrate an anomalou ituation, wherein uch a CPO allocation doe not exit, becaue the et of CPO and CGRO allocation coincide with each other. Example 2 (CP O = CGRO ) The econd example illutrate an anomalou ituation uch that the et of CPO and CGRO allocation coincide with each other. Conider the ame economy a Example 1 except for preference. Let U (c ) = c 1 + ω α ln c 2 α + ω β ln c 2 β for each S. 7 Thi lifetime utility function atifie the condition (b) of Propoition 1. In the current etting, one can eaily verify that, for every tationary feaible allocation c, it hold that U (c ) = c 1 + ω α ln( ω α c 1 α) + ω β ln( ω β c 1 β ) =: f (c 1 α, c 1 β ) for each S. Note that ( f / c 1 )(c 1 α, c 1 β ) = 1 ω /( ω c 1 ) < 0 and ( f / c 1 )(c1 α, c 1 β ) = ω /( ω c1 ) < 0 for any tationary feaible allocation c with 0 < c 1 < ω and any S. Therefore, the tationary feaible allocation x = {x 1, x 2 } defined by x 1 = 0 and x2 = ω for each (, ) S 0 S i the unique CGRO allocation. Note that x i alo the unique CPO allocation, and thu there i no CPO allocation that i not CGRO in thi economy. Thi example illutrate an anomalou ituation uch that the et of CPO allocation i too mall to be ditinguihed from the et of CGRO allocation. In thi example, uch a ituation i caued by quai-linearity of lifetime utility function. A hown in the following propoition, we can avoid ituation uch a thoe illutrated in Example 2 by impoing certain boundary condition on preference. Propoition 2 CP O CGRO but CP O CGRO, i.e., there exit a conditionally Pareto optimal allocation that i not conditionally golden rule optimal if, in addition to the condition in the previou propoition, lim c h1 0 U1 h(ch1, x h2 ) = for all x h2 R S ++ and all (h, ) H S hold. Proof of Propoition 2. See the Appendix. Q.E.D. Thi ection conclude with the following example of Propoition 2, demontrating the exitence of a CPO allocation that i not CGRO. Example 3 (CP O CGRO but CP O CGRO ) The third example illutrate a ituation uch that the et of CGRO allocation will be a proper ubet of the et of CPO allocation. 7 In thi and the next example, we conider that ln(0) to be well defined at. 6

10 Conider the ame economy a Example 1 except for preference. Let U (c ) = ω ln c 1 +c 2 α+c 2 β for each S. Thi lifetime utility function atifie both the condition (a) of Propoition 1 and the boundary condition of Propoition 2. Note that the tationary feaible allocation x = {x 1, x 2 } defined by x 1 = 0 and x2 = ω for each (, ) S 0 S i a CPO allocation. In the current etting, one can eaily verify that, for every tationary feaible allocation c, U (c ) = ω ln c 1 c 1 α c 1 β + ( ω α + ω β ) =: g (c 1 α, c 1 β ) for each S hold. Note that ( g / c 1 )(c 1 α, c 1 β ) = ω /c 1 1 > 0 and ( g / c 1 )(c1 α, c 1 β ) = 1 < 0 for any tationary feaible allocation c with 0 < c 1 < ω and any S. Therefore, the allocation x i not CGRO. In fact, one can verify that, for ufficiently mall ε > 0, the tationary feaible allocation y = {y 1, y 2 } defined by y 1 = ε and y2 = ω ε for each (, ) S 0 S i well-defined and CGRO-dominate x, becaue it improve the welfare of newly born agent. 4 Characterization of Optimality Criteria In the previou ection, we examined the relationhip between CPO and CGRO. It ha been hown that CGRO implie CPO under trict convexity of preference. Under uch condition, thi ection explore difference in the characterization of thee two optimality criteria. Thu, we trengthen the retriction on the economy by auming that U h i trictly quai-concave for all (h, ) H S in the ret of thi paper. Given an interior tationary feaible allocation c, let m h (c) = U h (ch )/U1 h(ch ) and let M h (c) = [m h (c)], S, where U1 h(ch ) = U h (c h )/ c h1 and U h (ch ) = U h (c h )/ c h2. The current retriction on preference imply that M h (c) i a poitive quare matrix. By the Perron- Frobeniu theorem, 8 any poitive quare matrix M ha a unique dominant root. Thi paper denote by λ f (M) the dominant root of a poitive quare matrix M. Thi tudy now characterize CPO and CGRO. The following reult extend Aiyagari and Peled (1991, Theorem 1) by characterizing the CPO of not an interior tationary equilibrium allocation but of an interior tationary feaible allocation. Theorem 1 An interior tationary feaible allocation c i conditionally Pareto optimal if and only if there exit a S S matrix M with poitive coefficient uch that ( h H) M = M h (c) 8 See, for example, Debreu and Hertein (1953) and Takayama (1974) for more detail on the Perron-Frobeniu theorem. 7

11 and it dominant root, λ f (M), i le than or equal to unity. Proof of Theorem 1. See the Appendix. Q.E.D. By Propoition 1 and Theorem 1, we now know that a CGRO allocation atifie λ f (M) 1 for an appropriate poitive quare matrix M. However, can we characterize CGRO by a harper condition than thi? The next reult give u more information on the characterization of CGRO allocation, i.e., a CGRO allocation atifie λ f (M) = 1 for ome appropriate poitive quare matrix M: Theorem 2 An interior tationary feaible allocation c i conditionally golden rule optimal if and only if there exit a S S matrix M with poitive coefficient uch that ( h H) M = M h (c) and it dominant root, λ f (M), i equal to unity. Proof of Theorem 2. See the Appendix. Q.E.D. Thi ection characterized optimality criteria of tationary feaible allocation by the dominant root of a matrix related to them in an economy with trictly convex preference. While the dominant root of a matrix related to a CGRO allocation mut be equal to one, wherea that to a CPO allocation i allowed to be le than one. Note that the dominant root criterion of CPO, not CGRO, provided in thi ection might not be applied to linear lifetime utility function. We conclude thi ection by preenting uch an example. Example 4 (Linear Preference) The fourth example illutrate an anomalou ituation uch that the dominant root criterion i inapplicable. Conider the ame economy a Example 1 except for preference. Let U (c ) = δc 1 + c 2 α + c 2 β for each S, where δ > 0. For every interior tationary feaible allocation c, one can eaily calculate the dominant root λ f of the matrix M(c) a λ f = 2/δ. We can thu obtain that λ f < 1 if δ > 2, λ f = 1 if δ = 2, and otherwie λ f > 1, where λ f i the dominant root of the matrix M(c). For δ (0, 2) (2, ), the dominant root criterion i applicable, i.e., all interior tationary feaible allocation are CPO if δ > 2 but are not CPO if δ < 2. However, for δ = 2, all interior tationary feaible allocation are CGRO, but not CPO, a hown in Example 1, although λ f = 1. 8

12 Strict quai-concavity of lifetime utility function, a aumed in thi ection, avoid the anomalou ituation illutrated in thi example. 5 Optimality of Stationary Equilibrium Allocation The previou ection characterized optimality criteria of tationary feaible allocation. Thee reult alo correpond to welfare analyi of tationary equilibrium. Thi ection examine the relationhip between optimality criteria and tationary equilibrium allocation. Thi ection define a tationary equilibrium uch that, in each ingle period, the one-period contingent claim market i complete: 9 Definition 3 A pair (Π, c) of a poitive price matrix Π = [π ], S of contingent commoditie and a tationary feaible allocation c = (c h ) (h,) H S i called a tationary equilibrium if for all (h, ) H S, c h belong to the et (x h1 arg max,x h2 ) R + R S + for all, S, h H (ch1 U h (x h ) : x h1 + S + ch2 ) = ω. x h2 π ωh1 + ω h2 π, S In thi definition, the former condition i the optimization problem of each agent (h, ) H S, and the latter i the market clearing condition. Let (Π, c) be a tationary equilibrium with c h 0 for all (h, ) H S, if any. Since, for all (h, ) H S, c h mut be a olution of the optimization problem of agent (h, ), it follow from the Kuhn-Tucker theorem that there exit ome λ h 0 uch that ( S) U h 1 (c h ) = U h (ch ) = U h c h1 (c h ) = λ h, U h c h2 (c h ) = λ h π, where λ h i the Lagrange multiplier of the lifetime budget contraint. Note that λ h > 0, becaue U h i trictly monotone increaing. Thu, we can oberve that [ ] U h ( h H) Π = [π ], S = (ch ) U1 h = M h (c). (ch ), S Therefore, the tationary equilibrium contingent claim price matrix Π can be alway repreented by the matrix of marginal rate of ubtitution at the tationary equilibrium allocation c. Thi 9 That i, we conider economy with equentially complete market. 9

13 obervation alo tate that all agent in the ame generation mut agree with the contingent claim price matrix at tationary equilibrium. The next two propoition follow immediately from the previou obervation and Theorem 1 and 2, repectively. Propoition 3 (Aiyagari and Peled,1991) For every tationary equilibrium (Π, c) with c h 0 for all (h, ) H S, c i conditionally Pareto optimal if and only if λ f (Π) 1. Propoition 4 For every tationary equilibrium (Π, c) with c h 0 for all (h, ) H S, c i conditionally golden rule optimal if and only if λ f (Π) = 1. Thee propoition characterize optimality of tationary equilibrium allocation. While the CPO of tationary equilibrium allocation i characterized by the dominant root of the contingent claim price matrix being le than or equal to one, their CGRO ha the dominant root exactly equal to one. Note that Propoition 3 extend Theorem 1 of Aiyagari and Peled (1991) by allowing poibilitie of heterogenou belief and noneparability of preference. 6 Welfare Theorem in Financial Economy We decribed the optimality of tationary equilibrium allocation in the previou ection. One of the important implication of the previou propoition i that tationary equilibrium allocation might be uboptimal even when market are equentially complete. It i known that an introduction of an infinitely lived outide aet uch a fiat money can remedy uch inefficiency. It i well known in the literature that an equilibrium with valued fiat money i CPO. In thi ection, we ue the previou analyi to how that a monetary equilibrium i actually CGRO, and to how that the firt and econd theorem of welfare economic hold if the criterion of optimality i CGRO. In order to introduce the poibility tranfer, needed for the econd welfare theorem, we introduce in addition to fiat money a mondatory unfunded ocial ecurity ytem. Mandatory unfunded ocial ecurity. We conider lump-um tranfer a a ocial ecurity ytem. Each young h H pay τ h1 at the current tate S. In contrat, each old h H born at tate S 0 receive τ h2 at the current tate S. It i aumed that the tranfer τ = {τ h1, τ h2 } h H atifie that τ1 h1 < ω h1 and τ h2 > ωh2 for all (, ) S 0 S. 10

14 It i alo aumed that the authority policy i balanced, i.e., h H τ h1 (, ) S 0 S. = h H τ h2 for all The financial aet. There exit an infinitely-lived outide aet, fiat money, which i available in poitive upply, i normalized to unity, and yield no dividend. A in the previou ection, we alo conider equentially complete contingent claim market. We then introduce a tationary monetary equilibrium: Definition 4 A triplet (q, Π, c) of a real price vector of money q R S ++, a nonnegative matrix Π = [π ], S of contingent claim, and a tationary feaible allocation c = (c h ) (h,) H S i called a tationary monetary equilibrium with tranfer τ if there exit money holding m R H S +, and portfolio of contingent claim [θ h ], S,h H (R S ) H S uch that for all (h, ) H S, (c h, m h, θ h ) belong to the et arg max (x h,z,ξ) (R + R S + ) R RS { U h (c) : x h1 ω h1 τ h1 q z S π ξ ( S) x h2 ωh2 + τ h2 + q z + ξ }, for all, S, h H (ch1 + ch2 ) = ω, h H mh = 1, and h H θh = 0. In thi definition, the former condition i the optimization problem of each agent (h, ) H S with equential budget contraint and the latter i the market clearing condition. Remark that, per Gottardi (1996), a tationary monetary equilibrium with tranfer exit generically and i locally iolated. 10 We now demontrate that the relationhip between tationary monetary equilibrium and CGRO i analogou to the fundamental theorem of welfare economic (welfare theorem) in the tandard Arrow-Debreu economy. Theorem 3 Every tationary monetary equilibrium with tranfer, if it exit and it allocation i interior, achieve conditional golden rule optimality. Thi theorem i an analogue of the firt welfare theorem. A mentioned above, it ha been well-known that a tationary monetary equilibrium allocation (with equentially complete market) i CPO. However, ince the definition of a tationary equilibrium doe not any pecial role 10 Further, for a tochatic overlapping generation economy, in which preference are additively eparable and H i ingleton, Ohtaki (2011) provide ufficient condition for the exitence and uniquene of a tationary monetary equilibrium. 11

15 to an initial date, o that the equilibrium i implicitly conidered on (, + ), the equilibrium allocation i actually CGRO. Note that there might exit a CPO allocation that cannot be implemented a a tationary monetary equilibrium, becaue tationary monetary equilibrium alway achieve CGRO. Therefore, the econd welfare theorem might not hold in tochatic OLG model if we adopt CPO a an optimality criterion. However, the following theorem how that the econd welfare theorem hold when we adopt CGRO, not CPO, a an optimality criterion. Theorem 4 Every interior conditionally golden rule optimal allocation, if any, can be achieved by a tationary monetary equilibrium with tranfer. Thi theorem i an analogue of the econd welfare theorem. A noted above, we might not be able to obtain thi theorem when we adopt CPO intead of CGRO a an optimality criterion. Theorem 4 ha an important implication, i.e., any interior CGRO allocation can be implemented a a tationary monetary equilibrium under an appropriate ocial ecurity ytem Concluion In a tochatic overlapping generation model, there exit two familiar criteria of optimality: conditional Pareto optimality (CPO) and conditional golden rule optimality (CGRO). Thi tudy ha examined the relationhip between thee two criteria. Contrary to a familiar intuition, thi tudy preent an example uch that the et of CPO allocation become a proper ubet of CGRO allocation. It ha been hown that uch an anomalou ituation i avoidable by auming trictly convex preference. The tudy ha alo examined how thee two concept are ditinguihed in their characterization under uch preference. We have hown that both thee criteria are characterized by condition on the dominant root of the agent common matrix of marginal rate of ubtitution. While CPO allow the dominant root of the matrix to be le than unity, CGRO require that it i exactly equal to unity, becaue CPO cope with an initial condition, wherea CGRO doe not. Thu, on the bai of their characteritic, we might ay that CGRO i tronger than CPO a a criterion of optimality. 11 Demange and Laroque (1999) alo provided welfare theorem imilar to our. However, while their reult were given in an economy with identical agent, our reult are hown in an economy with heterogenou agent. 12

16 It ha been known that a tationary monetary equilibrium achieve CPO. By applying our reult to welfare on tationary monetary equilibrium, we can conclude that a tationary monetary equilibrium achieve not only CPO but alo CGRO. Thi reult can be interpreted a the firt welfare theorem. Further, by adopting CGRO rather than CPO a an optimality criterion, thi tudy ha preented the econd a well a the firt welfare theorem in financial economy. Thee reult complement exiting reult for CPO. Appendix Proof of Propoition 1. Suppoe the contrary that there exit ome CGRO allocation c A that i not CPO. Becaue c i not CPO, there exit ome tationary feaible allocation b that CPO-dominate c, i.e., b atifie that ( (h, ) H S) b h2 0 ch2 0, U h (b h ) U h (c h ) with trict inequality omewhere. If b atifie that ( (h, ) H S) U h (b h ) > U h (c h ), then c i not CGRO and contradict the hypothei that c i alo. Thu, we aume without lo of generality that ( (k, τ) H S) b k2 0τ > c k2 0τ. We firt claim that b i1 j c i1 j for ome (i, j) H S. Suppoe the contrary that b h1 for all (h, ) H S. Becaue both c and b are tationary feaible allocation, it follow that = c h1 ( S) ω 0 h H b h2 0 = b h1 = c h1 = ω 0 b h2 0, h H h H h H which implie that h H bh2 0 = h H bh2. However, thi contradict the hypothei that bh2 c h2 0 0 for all (h, ) and bk2 0τ > ck2 0τ, which implie that h H bh2 0 > h H bh2. Therefore, bi1 j for ome (i, j) H S. 0 0 c i1 j We then claim that b i 2 j ci 2 j for ome (i, j ) H S. Suppoe the contrary that b h2 = c h2 for all (h, ) H S. Becaue both c and b are tationary feaible allocation, we obtain that, for all, S, h H (c h2 0 ch2 ) = ω 0 ω = (b h2 0 bh2 ), h H 13

17 which implie that ( S) becaue b h2 = c h2 h H c h2 0 = b h2 0, h H for all (h, ) H S. However, thi contradict the hypothei that b h2 0 ch2 0 for all (h, ) and b k2 0τ > ck2 0τ. Therefore, bi 2 j ci 2 j for ome (i, j ) H S. Now, let d := αc + (1 α)b for ome α (0, 1). It i a tationary feaible allocation, becaue h H d h1 + h H d h2 = α (c h1 + ch2 ) + (1 α) (b h1 + bh2 ) h H h H = ω for all (, ) S 0 S. It alo follow from quai-concavity of utility function that ( (h, ) H S) U h (d h ) U h (c h ) with trict inequality at either (i, j) or (i, j ), where the trict inequality follow from the retriction (a) and (b) on lifetime utility function, the fact that b i1 j c i1 j and b i 2 j ci 2 j, and the hypothei that U h (b h ) U h (c h ) for all (h, ) H S. Thi contradict the hypothei that c i a CGRO allocation. Q.E.D. Proof of Propoition 2. It i obviou that the tationary feaible allocation x = {x 1, x 2 } defined by x h1 = 0 and xh2 = ω for every (, ) S 0 S i well-defined and CPO, becaue any other tationary feaible allocation revie the initial old welfare for the wore. It i ufficient to how that thi x i not CGRO. However, the exitence of a tationary feaible allocation that CGROdominate x follow immediately from the given boundary condition that lim c h1 0 U1 h(ch1, c h2 ) = for all c h2 R S ++ and all (h, ) H S. Q.E.D. Proof of Theorem 1. Let c be an interior tationary feaible allocation. It i eay to verify that c i CPO if and only if there exit Pareto weight γ : H S R ++ and γ 0 : H S R + uch that c arg max b A (h,) H S Define the Lagrangian L by L = (h,) H S γ h U h (b h ) + ( ) γ h U h (c h ) + γ0 h c h2 0 γ0 h b h2 0. (h,) H S 14 (, ) S 0 S λ ω S(c h1 + ch2 ),

18 where λ i the Lagrange multiplier for the reource contraint. Note that the objective function i trictly quai-concave. Therefore, by Arrow and Enthoven (1961), the CPO of c can be completely characterized by the exitence of Pareto weight γ : H S R ++ and γ 0 : H S R + and Lagrange multiplier λ : S 0 S R +, which atify that ( (h, ) H S) γ h U h 1 (c h ) = S λ + λ 0, (1) ( (h, ) H S)( S) γ h U h (ch ) = λ, (2) ( (h, ) H S) γ h 0 λ 0 0 with equality if c h2 0 > 0. (3) Therefore, we hould claim equivalence between the exitence of γ, γ 0, and λ, atifying Eq.(1) (3), and the exitence of a poitive quare matrix M atifying that ( h H) M = M h (c) and λ f (M) 1. Thi tudy, however, omit the proof of thi claim becaue it i nearly identical to that of Theorem 1 of Aiyagari and Peled (1991). Q.E.D. Proof of Theorem 2. We firt claim that, for each interior tationary feaible allocation c and each S, there exit ome h H uch that c h 2 0 > 0. To verify thi claim, let c be an interior tationary feaible allocation and S. Becaue c i a tationary feaible allocation, we can obtain that, for all S, h H = ω = ω 0 c h1 c h1 + h H = + h H h H which implie that c h2 c h2 0, ( S) 0 < h H c h2 = c h2 0, h H where the firt inequality follow from the fact that c i interior. Therefore, for each S, there exit at leat one element h H uch that c h 2 0 > 0 becaue ch2 0 0 for every (h, ) H S. Let c be an interior tationary feaible allocation. It i eay to verify that c i a CGRO allocation if and only if there exit Pareto weight γ : H S R ++ uch that c arg max b A (h,) H S γ h U h (b h ). 15

19 Define the Lagrangian L by L = (h,) H S γ h U h (c h ) (, ) S 0 S λ ω S(c h1 + ch2 ), where λ i the Lagrange multiplier for the reource contraint. Note that the objective function i trictly quai-concave. Therefore, by Arrow and Enthoven (1961), the CGRO of c can be completely characterized by the exitence of Pareto weight γ : H S R ++ and Lagrange multiplier λ : S 0 S R + which atify that ( (h, ) H S) γ h U h 1 (c h ) = S λ + λ 0, (4) ( (h, ) H S)( S) γ h U h (ch ) = λ, (5) ( (h, ) H S) λ 0 0 with equality if c h2 0 > 0. (6) Note that, by the previou claim, we can treat λ 0 a zero for each S, becaue, for each S, there exit h H uch that c h 2 0 can ignore Eq.(6) and remove λ 0 from Eq.(4). > 0 and λ 0 i independent of index h. Therefore, we We hould now claim the equivalence between the exitence of γ and λ atifying Eq.(4) and (5) with λ 0 = 0 and the exitence of poitive quare matrix M atifying ( h H) M = M h (c) and λ f (M) = 1. Aume the exitence of γ and λ atifying Eq.(4) and (5) with λ 0 = 0 to how the exitence of poitive quare matrix M atifying λ f (M) = 1. Note that, by trict monotonicity of U h, m h (c) i poitive for all h and all, S. We can then obtain from Eq.(4) and (5) with λ 0 = 0 that ( h H)(, S) m h (c) = λ τ S λ, τ o that we can ignore the upercript h (and c) of m h (c), i.e., for all, S, there exit ome poitive number m uch that m = m h (c) for all h H. Then, it follow that (, S) λ = τ S λ τ m. Summing thi equation over S, we have (, S) α = αm, where α := τ S λ τ and M := [m ], S = M h (c) for all h H. Note that M i an S S matrix with poitive coefficient. Therefore, it follow from the Perron-Frobeniu theorem that λ f (M) = 1. 16

20 Aume now that the exitence of poitive quare matrix M atifying ( h H) M = M h (c) and λ f (M) = 1. Becaue M i an S S matrix with poitive coefficient, we can pick up the row eigenvector α 0 of M. Note that it atifie that α (I M) = 0, where I i the S S identity matrix. For all h H and all, S, define γ h and λ by γ h := α U1 h(ch ), λ := γ h U h (ch ). By their definition, we can obtain that (, S) λ = α U h (ch ) U h 1 (ch ) = α m, o that α = S λ for all S. It i now eay to verify that γ and λ atifie Eq.(4) and (5) with λ 0 = 0. Thi complete the proof. Q.E.D. Proof of Theorem 3. By the equential budget contraint of an agent, we can obtain the agent lifetime budget contraint uch that: for all (h, ) H S, d h1 + S d h2 π ωh1 τ h1 + (ω h2 + τ h )π + S π q q z. S By thi equation, we can obtain the no arbitrage condition when the money price i poitive, i.e., q = Π q for any tationary monetary equilibrium (q, Π, c) with tranfer. To how thi, we hould verify that ( S) q = S π q. Suppoe the contrary that q S π q for ome S. If q < S π q, then, for all h H, agent (h, ) will chooe a z and hi/her optimization problem ha no olution. However, if q > S π q, then, for all h H, agent (h, ) will chooe a z and hi/her optimization problem ha no olution. In any cae, we obtain a contradiction, o that q = S π q for all S. Let (q, Π, c) be a tationary monetary equilibrium (q, Π, c) with c h 0 for all (h, ) H S. We have obtained that Π q = q. Becaue q i now poitive for all S, it follow from the Perron-Frobeniu theorem that the S S matrix Π with poitive coefficient ha the dominant 17

21 root equal to unity. Thi complete the proof of Theorem 3. Q.E.D. Proof of Theorem 4. Let c = {c 1, c 2 } be an interior CGRO allocation. Becaue it i CGRO, there exit ome poitive matrix Π = [π ], S uch that Π = M h (c) for all h H and it dominant root i equal to one, i.e., λ f := λ f (Π) = 1. By the Perron-Frobeniu theorem, there exit a unique q R S ++ (up to normalization) uch that Π q = λ f q. Chooe the Euclidean norm of q to be mall enough and take any m = {m h } (h,) H S R H S ++, any τ = {τ h1, τ h2 } h H, and any θ = [θ h, ], S,h H to atify (a) the budget contraint in the firt period of each agent (h, ) H S: c h1 (b) ω h1 = ω h1 > τ h1 τ h1 q m h π θ h ; (7) S and τ h2 S; and (d) h H τ h1 > ωh2 for h H and, S; (c) h H mh = 1 and h H θh = 0 for = h H τ h2 for all (, ) S 0 S. 12 By their contruction, the firt order condition of all agent optimization problem at the tationary monetary equilibrium are atified. Q.E.D. Reference Aiyagari, S.R. and D. Peled (1991) Dominant root characterization of Pareto optimality and the exitence of optimal equilibria in tochatic overlapping generation model, Journal of Economic Theory 54, Arrow, K.J. and Alain C. Enthoven (1961) Quai-concave programming, Econometrica 29, Barbie, M., M. Hagedorn, and A. Kaul (2007) On the interaction between rik haring and capital accumulation in a tochatic OLG model with production, Journal of Economic Theory 137, Bloie, G. and F.L. Calciano (2008) A characterization of in efficiency in tochatic overlapping generation economie, Journal of Economic Theory 143, Chattopadhyay, S. (2001) The unit root property and optimality: a imple proof, Journal of Mathematical Economic 36, Let m h := 1/ H, τ h1 := ω h1 c h1 q / H, and θ h := 0 for all h H and all, S. By chooing the Euclidean norm of q to be ufficiently mall, m, τ 1, and θ can atify Eq.(7) and condition (a) (d) with ome {τ h2 } h H uch that τ h2 > ωh2 and h H τ h1 = h H τ h2 for (, ) S 0 S. 18

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