Hippolyte D'albis
email: hippolyte.d-albis@univ-tlse1.fr.

Demographic structure and capital accumulation

Keywords: JEL Classi…cation: D91, E13, J10 Continuous-time overlapping-generations models, Population aging

This paper develops an overlapping generations model to analyze the consequences of demographic structure changes induced by an exogenous shift in the birth rate. We …rst show that a …nite growth rate of the population that maximizes long-run capital per capita exists. Then, we examine the theoretical properties of this growth rate by showing that: (i) it corresponds to the demographic structure such that the average ages of capital holders and workers are equal; (ii) it is associated to an e¢ cient steady state; (iii) it increases with compulsory transfers from younger to older generations. Finally, we explain why standard overlapping generations models do not exhibit such a growth rate.

Introduction

Overlapping generations (OLG) models are the neoclassical literature's common tool for analyzing the economic consequences of demographic structure changes. In this paper, we focus on the impact of exogenous shifts in the birth rate on long-run capital accumulation. This relationship is crucial for analyzing the consequences of population aging on most macro-variables 1 , such as growth, assets prices and unemployment. An increase in the population growth rate simultaneously induces a reduction in capital per worker and an increase in savings; the problem is to …nd out which one has the strongest e¤ect. Standard OLG models with production in discrete or continuous time developed by Diamond [START_REF] Diamond | National debt in a neoclassical growth model[END_REF] and Blanchard [START_REF] Blanchard | Debt, de…cits and …nite horizons[END_REF], respectively, can be used to show that a birth-rate increase reduces long-run capital per capita. The economic intuition given to this result relies on the absence of intergenerational altruism: the size of future generations does not directly in ‡uence the saving choices of current generations. The reduction in aggregate consumption that follows a birth rate increase is therefore insu¢ cient for compensating the capital dilution e¤ect. Following Weil [START_REF] Weil | Overlapping families of in…nitely-lived agents[END_REF], the birth rate measures the degree of disconnection between generations, and newborn individuals are interpreted as "unloved children"or immigrants.

Empirical studies, however, do not show that demographic structure changes have a signi…cant impact of on capital accumulation or, more precisely, on its marginal productivity. In a recent paper, Poterba [START_REF] Poterba | Demographic structure and assets returns[END_REF] analyzes the historical relationship between population structure and assets return, including the interest rate. Using time series from the United States, Canada and the United Kingdom for the last seventy years. Poterba …nds no robust evidence of any impact of demographic variables on assets prices. At a …rst glance, these empirical …ndings seem to invalidate the OLG models while validating the Ramsey framework. In this framework 2 , where the population is composed of one unique, altruistic family, the demographic variables have no impact at the balanced equilibrium. That is because the optimal response to a birth-rate increase is to reduce consumption in order to keep capital per capita constant. However, in this paper, we argue that Poterba's results can be reproduced in an OLG model. We build a continuous time OLG model of individuals with …nite life-spans to show there exists a …nite population growth rate, or equivalently an age structure of the population, that maximizes capital per capita. The existence of this capital maximizer implies that (i) the sign of the impact of demographic growth on capital per capita is ambiguous; (ii) the two di¤erent demographic structures could be associated to the same capital per capita; (iii) and the demographic structure changes have a little impact on capital per capita when these structures are at the neighborhood of the one that maximizes capital per capita. Hence, we conjecture that the demographic structures of developed countries have remained close to the structure which maximizes capital per capita. Of course an estimation is required to prove this statement, which is not within the scope of this paper. Instead, we focus on the theoretical characterization of the demographic growth rate that maximizes steady-state capital per capita.

Our argument hinges on the simple but crucial assumption of a population composed of non-altruistic individuals who accumulate assets following a lifecycle behavior. This is standard in neoclassical literature, but is barely implemented in general equilibrium models, because of the technical di¢ culties that arise in the aggregation procedure. Thus, the two-period lifetime model by Diamond [START_REF] Diamond | National debt in a neoclassical growth model[END_REF] and the continuous time models by Blanchard [4] and Weil [START_REF] Weil | Overlapping families of in…nitely-lived agents[END_REF], which avoid these di¢ culties, are therefore extensively used in the literature. We show, however, that the assumptions that keep these models tractable, are responsible for the monotonic relationship they exhibit between population growth and capital per capita. We argue that it is possible to develop an analytically tractable model that reproduces the stylized fact highlighted by Poterba [START_REF] Poterba | Demographic structure and assets returns[END_REF]. We build a continuous time OLG model of individuals with …nite life-spans based on the pioneering works of Tobin [START_REF] Tobin | Life cycle saving and balanced growth[END_REF] and of Cass and Yaari [START_REF] Cass | Individual saving, aggregate capital accumulation, and e¢ cient growth[END_REF] 3 . We assume a general pattern of individual mortality, which includes such cases as the certain lifetime, the Blanchard's Poisson process and the Weil's in…nitely-lived families. This formalization makes the comparative discussion straightforward. Aggregation is made following Lotka's [START_REF] Lotka | Théorie analytique des associations biologiques[END_REF] assumption of a stable population structure as well as assumptions of perfect …nancial and insurance markets.

We then derive the following results. First, we give su¢ cient conditions for the existence and uniqueness of a steady-state path. These conditions, which are easy to interpret and compare with the existing OLG literature. We then perform a static comparative analysis and show that a …nite population growth rate that maximizes capital per capita exists. This result crucially depends on the …niteness of individual life-span. Moreover, the considered growth rate corresponds to the demographic structure, such that the average age of capital holders weighted by their wealth equals the average age of the workers weighted by their earnings. We demonstrate this result using the property of stable populations to exhibit easily averaged ages associated to individual variables 4 . Thus, in the steady-state equilibrium, the impact of demographic growth on capital per capita is positive if the average age of capital holders is lower than the average age of workers. By construction, this case cannot exist in the Diamond [START_REF] Diamond | National debt in a neoclassical growth model[END_REF] and Blanchard [START_REF] Blanchard | Debt, de…cits and …nite horizons[END_REF] models.

Before introducing the model, we want to emphasize that our analysis is positive. Indeed, our intention is di¤erent from the issue raised by Samuelson [START_REF] Samuelson | The optimum growth rate of a population[END_REF] on the optimal population growth rate. This growth rate is the one that maximizes welfare when the equilibrium is golden-rule. Instead, we make a static comparative analysis when the equilibrium is balanced. Nevertheless, we show that the equilibrium associated to the demographic growth rate that maximizes capital per capita, is e¢ cient. Moreover, we complete the analysis by introducing compulsory transfers between generations and show that transfers from younger to older generations increase the demographic growth rate that maximizes capital per capita.

The basic framework of the model is presented in Section 2 and the analysis of the e¤ect of a shift in the demographic growth rate on the steady-state capital per capita is made in Section 3. A general discussion of the results is proposed in Section 4. Section 5 concludes and the Appendix presents all the proofs of the propositions.

The model

This section presents an aggregation of individual consumption behavior under an uncertain lifetime in an overlapping generations model of neoclassical growth. It presents the steady state of such an economy and derives su¢ cient conditions for its existence and uniqueness. Time is continuous, and all the variables that depend on time are assumed to be continuous and di¤erentiable.

Individual behavior

We closely follow Yaari [START_REF] Yaari | Uncertain lifetime, life insurance and the theory of the consumer[END_REF] 5 . Individuals are uncertain about the length of their life. Let ( ) > 0 denote the probability density function of the random variable with support [0; ], where is the …nite maximum possible lifetime. Individuals also exhibit a pure discount factor ( ) that satis…es : (0; ] ! (0; 1) and (0) = 1; hence, the pure discount rate, denoted ( ) 0 ( ) = ( ) which may vary with age and might even be negative. Individuals only derive satisfaction from their consumption, denoted c ( ) 0 and have no bequest motives. Instantaneous utility is isoelastic with an elasticity of intertemporal substitution 2 (0; 1]. Hence, expected utility at birth of any individual is:

Z 0 ( ) 0 @ Z 0 (z) c (z) 1 1= 1 1 1= dz 1 A d (1)
During a lifetime, the labor supply is …xed and, w ( ) > 0; a given stream of labor income is received. There is a single asset and individuals have access to competitive capital and complete insurance markets. Individuals hence hold their entire wealth a ( ) in the form of actuarial notes that makes an insurance company their legatee. Since the notes are assumed actuarially fair, they yield the regular risk-free interest rate, r > 0; plus the hazard rate of death and are therefore the most lucrative asset if the individual wealth is positive. Moreover, the uncertainty about life-span makes the subscription to these notes the only way to borrow. Let p ( ) = R (z) dz denote the probability at birth that an individual will be alive at age . Hence, (z j ) = (z) =p ( ) is the conditional probability density evaluated at age z, given that the individual is alive at age , and ( j ) is the hazard rate of death at age . The hazard rate of death is supposed to be non-decreasing with respect to age: 0 ( j ) 0. The individual budget constraint hence writes:

a 0 ( ) = [r + ( j )] a ( ) + w ( ) c ( ) (2)
Individuals are born with no …nancial assets and face a terminal condition that forbids Ponzi games. The two following conditions therefore complete the individual program:

a (0) = 0 (3) a ( ) 0 (4)
We, moreover, assume that ( ), ( ), w ( ) and a ( ) are of class C 2 on [0; ] and that c ( ) is of call C 1 on [0; ]. For simplicity's sake, we de…ne G (:) and H (:) such that:

G (x) Z 0 p ( ) w ( ) exp ( x ) d (5)
H (x) Z 0 p ( ) [ ( )] exp ( x ) d (6)
which are both positive and decreasing. Note that R 0 p ( ) exp ( r ) d represents the expected discounted value of a constant ‡ow of one unit of output; this value is weighted with the income distribution in G (r) and with the preference parameters in H (r).

Given prices r and w, each individual chooses fc ( )g that maximizes (1) subject to (2), (3) and (4). The optimal behavior is now described by the following proposition:

Proposition 1 There exists a unique solution to the individual problem. For 2 [0; ], the optimal consumption pro…le satis…es:

c ( ) = G (r) H ((1 ) r) [ ( )] exp ( r ) (7)
while the optimal wealth satis…es:

a ( ) = Z p (z) p ( ) exp ( r (z )) [c (z) w (z)] dz (8)
with a ( ) = 0.

The individual behavior described by equation [START_REF] Buiter | Death, population growth and debt neutrality[END_REF] is the standard neoclassical one. Note that G (r) =H ((1 ) r) represents the product of the expected discounted ‡ow of earnings at birth and the marginal propensity to consume out of total individual wealth at birth, i.e. the initial consumption c (0). Deriving equation [START_REF] Buiter | Death, population growth and debt neutrality[END_REF] with respect to yields:

c 0 ( ) = [r ( )] c ( ) (9)
Perfect annuity markets imply that the optimal consumption growth rate only depends on the di¤erence between the interest rate and the pure discount rate.

Since the latter varies with age, individual consumption may then decrease during the life-span. Moreover, it may also be a concave function of age if the pure rate of discount su¢ ciently increases with age.

Concerning the optimal asset accumulation, its pro…le is characterized in the following proposition:

Proposition 2 For 2 [0; ], the optimal wealth is positive and hump-shaped if:

[r ( )] > 0 ( j ) ( j ) and 0 ( j ) r + ( j ) > w 0 ( ) w ( ) (10)
A borrowing period during the life-cycle is a possible output of the individual problem. However, under rather plausible assumptions, proposition 2 rules out this possibility. It is indeed su¢ cient to have a consumption growth rate greater than the hazard rate of death's growth rate and a bounded income growth rate. Remark that the hazard rate of death is usually assumed by demographers to follow a Gompetz' law such that ( j ) = exp (x ) with x > 0. Alternatively, if the individual consumption is decreasing during the life-span, the optimal wealth may not be hump-shaped. Finally, note that condition [START_REF] Cass | Individual saving, aggregate capital accumulation, and e¢ cient growth[END_REF] remains to be check at the equilibrium since it involves the interest rate that is to be an endogenous variable.

Remark 1. Our rather general speci…cations ease the comparison with OLG models developed by Blanchard [START_REF] Blanchard | Debt, de…cits and …nite horizons[END_REF] and Weil [START_REF] Weil | Overlapping families of in…nitely-lived agents[END_REF]. Blanchard assumes that an individual lifetime follows a Poisson process and that the pure discount rate and the wage stream are constants. With our notation, = 1, ( j ) = p; ( ) = , and w ( ) = w with p; ; w > 0. Thus, G (x) = w= (x + p) and exp ( (r + p) ) a ( ) 0. Weil interprets the individual as an in…nitely lived family, an assumption that can be obtained by setting p = 0. Replacing in equation ( 7), the individual consumption behavior becomes for all 0:

H (x) = 1= (x + p + ).
c ( ) = w [(1 ) r + p + ] (r + p) exp ( (r ) ) (11)
Simple manipulations of equation ( 11) yields the individual asset accumulation:

a ( ) = w (r + p) [exp ( (r ) ) 1] for 0 (12)
In Blanchard and Weil models, the consumption and asset accumulation pro-…les are hence a monotonic function of age.

Aggregation and competitive equilibrium

The demographic structure is based on Lotka's [START_REF] Lotka | Théorie analytique des associations biologiques[END_REF] stable population theory.

Each individual belongs to a large cohort of identical individuals. Therefore even though each individual's life-span is stochastic, there is no aggregate uncertainty. The law of large numbers is supposed to apply6 , and thus, the size of each cohort is decreasing at rate ( j ). Then, at time t, the size of the age-cohort is N (t ) p ( ) where > 0 is the birth rate and N (t ) is the size of the population at time t . Since a new cohort is born at each instant, N (t) satis…es:

N (t) = Z 0 N (t ) p ( ) d (13)
Lotka demonstrates that the demographic structure (i.e. the relative size of each cohort in the population) reaches a steady state when the birth and death rates have been constant over a su¢ ciently long period of time. The stationary distribution, characterized by the demographic growth rate n, is obtained by replacing N (t) = N (0) exp (nt) in ( 13) and then solving:

Z 0 p ( ) exp ( n ) d = 1 (14)
The unique real solution of equality (14) de…nes7 the population growth rate n whose sign depends on the di¤erence between the birth rate and the inverse of the life expectancy at birth. Formally, n 0 if and only if 1=

R 0 p ( ) d .
Note that the relative size of each cohort in the population, p ( ) exp ( n ), is then time independent, which is why the population is said to be stable.

Aggregate wage per capita, denoted as w, satis…es:

w Z 0 p ( ) exp ( n ) w ( ) d (15)
while consumption per capita, denoted as c, satis…es:

c Z 0 p ( ) exp ( n ) c ( ) d (16)
Hence, replace equation ( 7) in ( 16) and replace using ( 5) and ( 15), to obtain:

c = w G (r) G (n) H (n r) H ((1 ) r) (17)
Moreover, when the individuals consumption pro…le is optimal, aggregate …nancial wealth per capita, denoted as a, satis…es8 :

a = c w r n (18)
There is a unique material good, whose price is normalized to 1. It can be used for consumption or for adding to the capital stock. This good is produced by many competitive …rms whose aggregate activity is described by a constant return-to-scale production function with labor and capital as inputs. Let k be the capital per capita, 0 be the depreciation rate and f (:) be the production function in intensive form. Assume that f : [0; 1) ! [0; 1); that f 0 (k) > 0 and f 00 (k) < 0 for all k > 0, and that lim

k!0 f 0 (k) = 1. Factor prices equal marginal products; hence w = f (k) kf 0 (k) and r = f 0 (k) .
As in Diamond [START_REF] Diamond | National debt in a neoclassical growth model[END_REF] and Blanchard [START_REF] Blanchard | Debt, de…cits and …nite horizons[END_REF], the equilibrium condition is obtained by identifying asset holdings with capital stock: a = k. The analysis of steadystate equilibrium turns out to be a …xed-point problem, k = (k), where function is obtained by …rst replacing equation ( 17) in ( 18) and then replacing factor prices such that:

(k) f (k) kf 0 (k) f 0 (k) (n + ) " G (f 0 (k) ) G (n) H (n (f 0 (k) )) H ((1 ) (f 0 (k) )) 1 # (19) Let us introduce s (k) kf 0 (k) =f (k), the share of capital in output and " (k) f 0 (k) [1 s (k)] = [kf 00 (k)],
the elasticity of substitution between capital and labor. The concavity of f implies that s 2 (0; 1) and " > 0 for all k > 0.

Proposition 3 There exists k > 0 such that k = k if lim k!0 [ kf 00 (k)] > 1 (20)
k is unique if Our existence condition ( 20) is close to the condition proposed by Bommier and Lee [START_REF] Bommier | Overlapping generations models with realistic demography: statics and dynamics[END_REF] who extend to a continuous-time framework the …ndings of Konishi and Perera-Tallo [START_REF] Konishi | Existence of steady-state equilibrium in an overlapping-generations model with production[END_REF] for the Diamond model [START_REF] Diamond | National debt in a neoclassical growth model[END_REF]. Their condition, named the "non-vanishing labor share", has the advantage of relying only on the production function. Formally it can be written as:

s (k) " (k) and d dk s (k) " (k) ! 0 (21)
lim k!0 s (k) 2 [0; 1) (22)
However, ( 22) is more restrictive than [START_REF] Grandmont | Money and Value: a Reconsideration of Classical and Neoclassical Monetary Theories[END_REF]. To demonstrate that point, let us observe that

lim k!0 " 1 s (k) s (k) f 0 (k) # lim k!0 [ kf 00 (k)] (23)
Let us now recall the Inada condition on f 0 (k) and conclude that [START_REF] Kehoe | Multiplicity of equilibria and comparative statics[END_REF] implies

lim k!0 [ kf 00 (k)] = +1.
In condition [START_REF] Grandmont | Money and Value: a Reconsideration of Classical and Neoclassical Monetary Theories[END_REF], the key elements are the maximum possible lifetime and the elasticity of inter-temporal substitution. Strong income e¤ects yielded by a small , may hence be compensated by a longer lifespan. Moreover, in Blanchard [START_REF] Blanchard | Debt, de…cits and …nite horizons[END_REF] and Weil's [START_REF] Weil | Overlapping families of in…nitely-lived agents[END_REF] models, where = 1, a competitive equilibrium always exists.

Multiple equilibria may occur in such a framework: see notably Kehoe [START_REF] Kehoe | Multiplicity of equilibria and comparative statics[END_REF] for production economies and Ghiglino and Tvede [START_REF] Ghiglino | No-trade and uniqueness of steady states[END_REF] for economies with heterogeneous endowments. Further restrictions on the production function are hence needed to exclude this possibility. Our conditions [START_REF] Judd | The law of large numbers with a continuum of i.i.d. random variables[END_REF] state that inputs should be rather substitutable, especially if capital per capita is low at equilibrium. These conditions are notably satis…ed by production functions with a constant elasticity of substitution that is greater than one (" 1); that is the functions that verify the Inada condition for k = 0. Let us also note that relaxing the assumption of 2 (0; 1] may produce multiple equilibria.

Remark 2. The complete analysis of the inter-temporal equilibrium is beyond the scope of this paper. Note nevertheless, that recent contributions have pointed out the importance of the initial condition of the economy on the existence of the equilibrium. Burke [START_REF] Burke | Equilibrium for overlapping generations in continuous time[END_REF] notably extend to continuous time the result that the equilibrium may fail to exist if time has a …nite starting point (See Geanakoplos and Polemarchakis [START_REF] Geanakoplos | Overlapping generations[END_REF]). As a consequence, the particular solution that is given by the steady-state equilibrium may also fail to exist. Moreover, the dynamics of the equilibrium path is not necessarily monotoneous: in a pure exchange economy, Demichelis and Polemarchakis [START_REF] Demichelis | Frequency of trade and the determinacy of equilibrium paths: logarithmic economies of overlapping generations under certainty[END_REF] show the existence of exponnentially decreasing ‡uctuations along the transition while Ghiglino and Tvede [START_REF] Ghiglino | Dynamics in OG economies[END_REF] show that a cycle may exist.

A graphical illustration. We propose a graphical example of the equilibrium in the (k; c) space when conditions ( 20) and ( 21) are respected. Note …rst that equation ( 18) may be written as c = (r n) a + w; at the equilibrium, this de…nes c 1 (:) such that:

c 1 (k) = f (k) (n + ) k (24)
The c 1 (k) line represents the resource constraint of the economy. If n + > 0, c 1 (k) is hump-shaped with golden-rule consumption that occurs at the point where c 1 (k) reaches its maximum. Then, we use the de…nition of the wages given above and replace it in [START_REF] Gale | Pure exchange equilibrium of dynamic economic models[END_REF]; at the equilibrium, this de…nes c 2 (:) such that:

c 2 (k) = k [f 0 (k) (n + )] 1 G(n) G(f 0 (k) ) H((1 )(f 0 (k) )) H(n (f 0 (k) )) (25)
The c 2 (k) line stands for the aggregation of individual consumption decisions for each level of capital. Here, for example, …gure 1 represents 9 the situation where n + > 0, f (0) = 0 and there is dynamic e¢ ciency. 9 Details of the derivation of this …gure are given in the Appendix. This …gure is the extension, for any stationary demographic structure, of the one presented in Blanchard [START_REF] Blanchard | Debt, de…cits and …nite horizons[END_REF].

Demographic growth and capital per capita

This section analyses the impact of a shift in the population growth rate on the steady-state capital per capita and shows that a …nite growth rate that maximizes capital per capita exists. It then characterizes this capital maximizer.

Main results

It is useful to de…ne x , the average age calculated cross-sectionally such that:

x R 0 p ( ) exp ( n ) x ( ) d R 0 p ( ) exp ( n ) x ( ) d (26)
where x ( ) is a relevant characteristic. In the case x ( ) = 1; 1 is the standard de…nition of the average age of the population. If x ( ) = w ( ) ; w is the average age of workers weighted by their earnings and equivalently, if x ( ) = a ( ) ; a is the average age of capital holders weighted by their wealth.

We now implicitly consider that the change in the demographic growth is induced by a change in the birth rate. It is, however, not worth representing changes in the birth rate since its long-run relationship with demographic growth is monotonously increasing. Indeed, implicit di¤erentiation of condition ( 14) yields:

dn d = 1 1 > 0 (27)
Hence, a static comparative of demographic growth on steady-state capital per capita yields the following result:

Proposition 4 The impact of an increase in population growth on capital per capita depends on the di¤erence between the average age of capital holders and the average age of workers; one indeed has:

d k dn = k 1 0 k [ a w ] (28)
Proposition 4 states that an increase in population growth does not necessarily reduce the steady-state capital per capita but depends on income and population distributions. Observe moreover that a is an endogenous variable while w is exogenous. The interpretation of this result will be given after the next proposition.

Proposition 5 There exists a …nite value of the population growth rate, denoted as n , that maximizes steady-state capital per capita.

Propositions 4 and 5 show that the relationship between the population growth rate and steady-state capital per capita is non monotonic and that capital per capita reaches a maximum when the average age of capital holders equals the average age of workers. The intuition of this result can be obtained by comparing the impact of demographic growth on the age structure of the working population and the pattern of individual assets. Let us …rst observe that the average age of the workers is decreasing with n; at the limit, it equals 0 for n = +1 and for n = 1. Next, let us recall that the optimal asset accumulation pattern during lifetime is such that a (0) = a ( ) = 0. Therefore, when n goes to plus or minus in…nity, (i.e. when 1 goes to 0 or ) the capital accumulated in the economy is very low. Conversely, the steady-state capital per capita is maximal when the distribution of population concentrates a large mass of individuals around the maximum wealth in the life-cycle accumulation pro…le. Remark that Proposition 5 does not rule out multiples n . Observe nevertheless, that if n is unique, the relationship between n and k is humpshaped.

Remark 3. An n-maximizer has also been recently presented in Boucekkine, de la Croix and Licandro [START_REF] Boucekkine | Vintage human capital, demographic trends and endogenous growth[END_REF]. In an endogenous growth model with schooling and retirement choices, they …nd a hump-shaped relationship between the population growth rate and the per-capita growth rate of human capital. Their result is based on a functional but realistic survival law and an instantaneous utility that is linear with respect to consumption. There is no capital in their model, but the existence of an n-maximizer relies on the same reasoning: the vintage structure of human capital.

Optimality

The population growth rate that maximizes capital per capita is not optimal since it does not maximize individual welfare or aggregate consumption; it can be only stated that the equilibrium to which it is associated with is e¢ cient.

To see that point, let us …rst consider the following proposition:

Proposition 6 Let k be such that k = 0 and such that

f 0 k = + . Then, n < (29)
Proposition 6 introduces , the lower bound on the interest rate, associated to k, the maximal value for capital per capita at equilibrium. This bound corresponds to the interest rate above which outstanding debt in the economy outweighs positive assets. If the individual pure discount rate is constant, such that ( ) = , then = . Otherwise, depends on the model parameters, including demographic ones. Thus, proposition 5 states that the population growth rate that maximizes capital per capita is lower than the lower bound on the interest rate. This result is quite intuitive if the individual consumption pro…le is supposed to be increasing: then, the age at which individuals have a maximal wealth takes place during the second half of their lives. Therefore, capital per capita is maximized by a demographic structure with a relatively high average population age, which means a rather low population growth rate.

Corollary 1. At the steady-state equilibrium associated with n , there is under-accumulation of capital.

The de…nition of , given in proposition 6, implies that any steady-state k sat-is…es f 0 k > . Consequently, the equilibrium associated with n exhibits an interest rate strictly greater that the population growth rate. Consequently, the equilibrium (i) cannot be a golden-rule equilibrium and (ii) is e¢ cient as proved by de La Croix and Michel [START_REF] De La Croix | A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations[END_REF] 10 . This does not implies that the equilibrium associated with n is Pareto optimal. This problem has been …rst discussed in Cass and Yaari [START_REF] Cass | Individual saving, aggregate capital accumulation, and e¢ cient growth[END_REF] and carefully studied by Grandmont [START_REF] Grandmont | Money and Value: a Reconsideration of Classical and Neoclassical Monetary Theories[END_REF],

Wang [START_REF] Wang | Money, competitive e¢ ciency, and intergenerational transactions[END_REF], and Duc and Ghiglino [START_REF] Duc | Optimality of barter steady states[END_REF]. The latter notably propose a general characterization of the endowment distributions leading to an optimal barter steady-state.

Remark 4. The aggregate consumption per capita is not maximized when the population growth rate equals n . Our work then di¤er from those of Samuelson [START_REF] Samuelson | The optimum growth rate of a population[END_REF] and Michel and Pestieau [START_REF] Michel | Population growth and optimality: when does serendipity hold?[END_REF] who have studied the existence of an "optimal" population growth rate that maximize individual's welfare when the economy is at the golden-rule equilibrium. In a continuous-time OLG framework, Arthur and McNicoll [START_REF] Arthur | Samuelson, population and intergenerational transfers[END_REF], Lee [START_REF] Lee | Age structure, intergenerational transfers and economic growth: an overview[END_REF] and Willis [START_REF] Willis | Life cycle, institutions, and population growth: a theory of the equilibrium interest rate in an overlapping generation model[END_REF] interpret this "optimal"population growth rate in terms of average ages of consumption and earnings.

Intergenerational transfers

Real world economies include many kinds of non-competitive intergenerational transfers such as child rearing, Pay-As-You-Go pension systems and intrafamilial transfers. In this subsection, we will introduce a simple intergenerational transfer scheme in our model and analyze its impact on n .

Following Blanchard [START_REF] Blanchard | Debt, de…cits and …nite horizons[END_REF], we introduce a simple intergenerational transfers scheme assuming that disposable incomes are growing with age at the constant rate of : a > 0 (respectively, a < 0) implies a transfer scheme from younger to older (from older to younger) individuals. Let us assume that the disposable income, denoted w ( ), is such that:

w ( ) = w ( ) exp ( ) (30)
where is a constant whose value is determined to ensure that the following equality is satis…ed:

Z 0 p ( ) w ( ) exp ( n ) d = Z 0 p ( ) w ( ) exp ( n ) d (31)
Replacing (30) yields = G (n) =G (n ) and therefore:

w ( ) = w ( ) G (n) G (n ) exp ( ) (32)
Then, equation ( 7) can be rewritten as:

c ( ) = G (n) G (n ) G (r ) H ((1 ) r) [ ( )] exp ( r ) (33)
and equation [START_REF] Ghiglino | Dynamics in OG economies[END_REF] becomes:

(k) f (k) kf 0 (k) f 0 (k) (n + ) " G (f 0 (k) ) G (n ) H (n (f 0 (k) )) H ((1 ) (f 0 (k) )) 1 # (34)
It is easy to show that, upon conditions stated in proposition 3, the existence and uniqueness of a steady-state equilibrium are guaranteed for any …nite . The impact of on k is negative, which is standard: increasing transfers to older individuals reduces private savings and therefore steady-state capital per capita. We will discuss the impact of on n in the following proposition.

Proposition 7 Let n be the demographic growth rate that maximizes k, given by k = k; n; , and suppose n is unique; thus:

dn d > 0 (35)
Proposition 7 states that transfers from younger to older (respectively, from older to younger) individuals increase (reduce) the demographic growth rate that maximizes capital per capita.

To interpret proposition 7, it is necessary to recall proposition 4 and observe that the impact of on n depends on two opposite e¤ects. On the one hand, w increases with : equation [START_REF] Seierstad | Optimal control theory with economic applications[END_REF] implies that transfers to older individuals reduce the wage of newborn individuals and increase the wage growth rate. On the other hand, a also increases with . This is the consequence of the induced increase in the interest rate that postpones the age at which individuals stop saving. Hence, proposition 7 states that the …rst e¤ect dominates the second one, and therefore that the introduction of transfers to older individuals increases n . This result is also supported by analyzing the impact of on . Simple calculus shows that the impact is positive, which enhances the idea of proposition 7.

Comparison with the literature

We will now explain why commonly used OLG models developed by Diamond [START_REF] Diamond | National debt in a neoclassical growth model[END_REF] and Blanchard [START_REF] Blanchard | Debt, de…cits and …nite horizons[END_REF] imply a negative relationship between population growth and capital per capita.

Diamond [START_REF] Diamond | National debt in a neoclassical growth model[END_REF] supposes each individual lifetime to be composed of two discrete periods. During the …rst period, individuals work and earn a wage that allows them to save for retirement. To do so, they buy the assets accumulated by those in the second period of their life. This assumption eases aggregation because the savings of the younger individuals correspond to the total capital of the economy. However, by construction, elder individuals hold all the assets. Thus, the average age of capital holders is always greater than the average age of workers. It is true that in our model, individuals work their entire lives. Such an assumption would not change the implication of the two-period model: if individuals were allowed to work during both periods, the average age of earnings would be equal to the average age of the population, but would still be lower than the average age of capital holders 11 . The monotonic result of the Diamond model therefore comes from the exogenous length of time that the individual saves. Conversely, if a lifetime covers N 3 discrete periods as in Gale [START_REF] Gale | Pure exchange equilibrium of dynamic economic models[END_REF] or a …nite interval as in our framework, the "economic youth", de…ned as the saving period, is endogenous and the result of proposition 5 applies.

The continuous time model of Blanchard [4] is also widely used in macroeconomics. We will now discuss Buiter [START_REF] Buiter | Death, population growth and debt neutrality[END_REF] and Weil's [START_REF] Weil | Overlapping families of in…nitely-lived agents[END_REF] models; these two extensions of Blanchard allow for changes in the population growth rate. Buiter assumes a non-stationary population that grows at rate n = p, where is the birth rate and p is the death rate de…ned in remark 1. The identi…cation of the aggregate with the private death rate implies that the average age of the population, and consequently the average age of workers when the wage is age-independent, depends only on the birth rate: w = 1= . To obtain Weil speci…cations, we only need to assume that p = 0. In that case, the population growth rate is equal to the birth rate. Because cohorts have homogenous horizons whatever their age, human wealth and propensity to consume out total wealth are age-independent. Moreover, as mention in remark 1, the consumption and asset accumulations pro…les are monotonic functions of age. Now let us compute [START_REF] Demichelis | Frequency of trade and the determinacy of equilibrium paths: logarithmic economies of overlapping generations under certainty[END_REF] in [START_REF] Lee | Age structure, intergenerational transfers and economic growth: an overview[END_REF] to obtain the average age of capital holders:

a = (2 (r )) ( (r )) for r 2 ; + ! (36)
Hence, a > w . This result hinges on the assumption that individuals have age-independent horizons. Every agent therefore behaves as a newborn and accumulates assets forever: a ( ) is not hump-shaped but increasing and convex. On the other hand the relative size of a cohort, given by exp ( ), is always decreasing even if the demographic growth rate is negative. Therefore, Blanchard-Buiter and Weil's models systematically exhibit an average age of capital ownership greater than the average age of the population.

Conclusion

In this paper, we have developed a continuous time, overlapping generations model to analyze the impact of demographic changes on steady-state capital per capita. First, we presented new conditions for the existence and the uniqueness of a steady state with a general pattern of individual mortality and income distribution. We then performed a comparative static analysis of the impact of the population growth rate on capital per capita, and showed that the functional relationship between those two variables is non-monotonic. Models by Diamond [START_REF] Diamond | National debt in a neoclassical growth model[END_REF] and Blanchard [START_REF] Blanchard | Debt, de…cits and …nite horizons[END_REF], therefore, constitute particular exceptions by exhibiting a strictly decreasing relationship. We next de…ned an age structure of the population that maximizes capital per capita. This structure made the average ages of capital holders and workers equal. Finally, we showed that the growth rate of the population that corresponds to the considered structure is lower than the equilibrium interest rate and increases with non-competitive transfers to older individuals.

Our model constitutes a rather simple framework for the analysis of agedependant behaviors. Notably, consequences of fertility and labor-market decisions can be directly investigated in such a general equilibrium setting. Moreover, a study of transition dynamics should be an interesting extension of this paper since the dynamics are not likely to be monotonic. Lotka [START_REF] Lotka | Théorie analytique des associations biologiques[END_REF] demonstrates that any population with constant birth and death rates converges toward a stable age structure with exponentially decreasing oscillations. The problem is similar in economic models, although they deal with a forward variable. Numerical methods are proposed in Laitner [START_REF] Laitner | The dynamic analysis of continuous-time life-cycle savings growth models[END_REF] and Boucekkine et al. [START_REF] Boucekkine | Vintage human capital, demographic trends and endogenous growth[END_REF] to analyze the behavior of non-stationary paths. The latter shows that the vintage structure of the model creates discrete delays in the dynamic behavior of aggregate variables which are then governed by echo e¤ects. 12 ; change …rst the order of integration of function [START_REF] Abel | The e¤ects of a baby boom on stock prices and capital accumulation in the presence of social security[END_REF] to rewrite the objective as

Appendix PROOF. [Proof of proposition 1] The individual program can be solve using classical calculus of variation

Z 0 p ( ) ( ) u (c ( )) d (37)
with

u (c ( )) = c ( ) 1 1= 1 1 1= (38)
Using ( 2), ( 3) and ( 4), the program becomes

max Z 0 L ( ; a; a 0 ) d s:t: a (0) = 0 a ( ) 0 h r p 0 ( ) p( ) i a ( ) + w ( ) a 0 ( ) 0 (39)
with

L ( ; a; a 0 ) = p ( ) ( ) u " r p 0 ( ) p ( ) # a ( ) + w ( ) a 0 ( ) ! ( 40
)
Since L is a C 2 -function of ( ; a; a 0 ), necessary conditions for a C 2 -function a ( ) to be a solution of program (39) are the Euler equation:

d d " p ( ) ( ) u 0 " r p 0 ( ) p ( ) # a ( ) + w ( ) a 0 ( ) !# = " r p 0 ( ) p ( ) # p ( ) ( ) u 0 " r p 0 ( ) p ( ) # a ( ) + w ( ) a 0 ( ) ! (41)
and the terminal condition:

@L ( ; a; a 0 ) @a 0 ! = 0 (42)
Equation (41) rewrites using (38) as follows

c 0 ( ) = " r + 0 ( ) ( ) # c ( ) (43)
which yields

c ( ) = c (0) [ ( )] exp ( r ) (44)
Now observe that @L ( ; a; a 0 ) @a 0 = ( ) [p ( )] 1+ 1 (45)

([rp ( ) p 0 ( )] a ( ) + p ( ) w ( ) p ( ) a 0 ( ))
1

Hence for all a ( ) 0, the terminal condition (42) is satis…ed. We, nevertheless, now show that there exists a unique candidate for optimality. To do so, integrate forward in time condition (2) to obtain

a ( ) = Z p (z) p ( ) exp ( r (z )) [c (z) w (z)] dz (46)
When = 0 use condition (3) to derive the intertemporal budget constraint at birth

Z 0 p ( ) exp ( r ) c ( ) d = Z 0 p ( ) w ( ) exp ( r ) d (47)
and replace (44) to obtain the individual initial consumption

c (0) = R 0 p ( ) w ( ) exp ( r ) d R 0 p ( ) [ ( )] exp ( (1 ) r ) d (48)
Consequently, there exists a unique consumption pro…le that satis…es the necessary condition. Replacing this pro…le in (46) yields a unique wealth pro…le. Since for all z 2 ( ; ) one has p (z) < p ( ) observe …nally with (46) that a ( ) = 0.

We conclude this proof showing that the necessary conditions are su¢ cient.

To do so, we have to prove that for all 2 [0; ] ; L ( ; a; a 0 ) is concave as a function of (a; a 0 ). The Hessian matrix writes

H = 2 6 4 p ( ) ( ) h r p 0 ( ) p( ) i 2 u 00 (c ( )) p ( ) ( ) h r p 0 ( ) p( ) i u 00 (c ( )) p ( ) ( ) h r p 0 ( ) p( ) i u 00 (c ( )) p ( ) ( ) u 00 (c ( )) 3 7 5
(49) Then, L is concave and the unique solution of the Euler equation is globally maximal.

PROOF. [Proof of proposition 2] Observe …rst that since the optimal wealth pro…les satis…es a (0) = a ( ) = 0, there exists at least one age denoted ^ 2 (0; ) such that a 0 (^ ) = 0. We proceed by showing that under [START_REF] Cass | Individual saving, aggregate capital accumulation, and e¢ cient growth[END_REF] one has a 00 (^ ) < 0. Deriving equation ( 2) with respect to , yields

a 00 (^ ) = 0 (^ j^ ) a (^ ) + w 0 (^ ) c 0 (^ ) (50)
Then, a 00 (^ ) < 0 is equivalent to

c 0 (^ ) > 0 (^ j^ ) r + (^ j^ ) c (^ ) w (^ ) " 0 (^ j^ ) r + (^ j^ ) w 0 (^ ) w (^ ) # (51)
Then if one assumes 0 ( j ) = [r + ( j )] > w 0 ( ) =w ( ), it is su¢ cient that c 0 ( ) =c ( ) > 0 ( j ) = ( j ). Use [START_REF] Cass | Optimum growth in an aggregative model of capital accumulation[END_REF] to obtain [START_REF] Cass | Individual saving, aggregate capital accumulation, and e¢ cient growth[END_REF]. Consequently, (i) wealth is positive for 2 (0; ) (ii) there is a unique ^ .

Derivation of equation [START_REF] Ghiglino | No-trade and uniqueness of steady states[END_REF] Recall that the optimal wealth accumulation at age is such that

a ( ) = Z exp Z z r + (u ju ) du [c (z) w (z)] dz (52)
The de…nition of aggregate wealth per capita is

a Z 0 p ( ) exp ( n ) a ( ) d (53)
where p ( ) = exp ( R 0 (u ju ) du). Replacing (52) in a and rearranging yields

a = Z 0 exp ((r n) ) ( Z p (z) exp ( rz) [c (z) w (z)] dz ) d (54)
which, changing the order of integration, turns to be equal to

a = Z 0 p ( ) exp ( r ) [c ( ) w ( )] Z 0 exp ((r n) z) dz d (55)
which yields

a = R 0 p ( ) exp ( n ) [c ( ) w ( )] d r n R 0 p ( ) exp ( r ) [c ( ) w ( )] d r n (56)
Using (47), and then ( 16) and [START_REF] Duc | Optimality of barter steady states[END_REF], equation ( 18) follows.

Now it is useful for the next proofs to consider the following lemma:

Lemma 8 Let function J : R ! R ++ be such that

J (x) G (x) G (n) H (n x) H ((1 ) x) (57) (i) J is strictly convex, (ii) lim x!+1 J (x) = lim x! 1 J (x) = +1.
PROOF. As a preliminary, consider the function J …rst derivative

J 0 (x) = [ h (n x) + (1 ) h ((1 ) x) g (x)] J (x) (58)
where

g (x) G 0 (x) G (x) = R 0 p ( ) exp ( x ) d R 0 p ( ) exp ( x ) d > 0 (59)
and

h (x) H 0 (x) H (x) = R 0 p ( ) [ ( )] exp ( x ) d R 0 p ( ) [ ( )] exp ( x ) d > 0 (60)
The function J second derivative is

J 00 (x) = h 2 h 0 (n x) + (1 ) 2 h 0 ((1 ) x) g 0 (x) i J (x) + [ h (n x) + (1 ) h ((1 ) x) g (x)] 2 J (x) (61)
(i) To prove that J 00 (x) > 0 we show that g 0 < 0 and h 0 < 0 and that (1 ) 2 h 0 ((1 ) x) > g 0 (x). Observe that g 0 (x) has the sign of [G 0 (x)] 2 G 00 (x) G (x). Let functions u and v be such that u ( ; x) = [exp ( x ) p ( )] 1=2 and v ( ; x) = u ( ; x) : Then G 0 (x) = R 0 u ( ; x) v ( ; x) d ; using the Cauchy-Schwartz inequality, g 0 (x) < 0. Similarly h 0 (x) < 0. Now, since lim !1

(1 ) 2 h 0 ((1 ) x) = 0 and since (1 ) 2 h 0 ((1 ) x) is increasing in , for all 2 (0; 1] ; part (i) of the lemma follows.

(ii) Observe that lim

x!+1 g (x) = lim x!+1 h (x) = 0 and lim x! 1 g (x) = lim x! 1 h (x) = . Therefore, lim x!+1 J 0 (x) J (x) = lim x! 1 J 0 (x) J (x) = (62)
and part (ii) of the lemma follows.

PROOF. [Proof of proposition 3] We derive conditions for (i) existence and (ii) uniqueness of a balanced steady state k that solves k = k , where function , which is given by [START_REF] Ghiglino | Dynamics in OG economies[END_REF], can be rewritten for convenience as follows

(k) f (k) kf 0 (k) f 0 (k) (n + ) [J (f 0 (k) ) 1] (63)
with J given by (57).

(i) Existence. We proceed in two steps: we …rst show it exists a unique k > 0 satisfying k = 0, and that (k) is continuous and positive for all k 2 0; k ; then, we propose a condition such that lim

k!0 (k) > 0.
Step 1. To prove the existence of k, we study the solutions of equation (k) = 0 for all k > 0. Given the concavity of f for all k > 0, one has f (k) kf 0 (k) > 0.

Hence, it is su¢ cient to look at the solutions of

J (f 0 (k) ) 1 f 0 (k) (n + ) = 0 (64)
Let us de…ne the 'golden rule'level of capital per capita, k gr such that

f 0 (k gr ) = n + (65)
Observe with (57) that J (f 0 (k gr ) ) = 1. Now, applying l'Hôpital's rule, using (58) when x = n, yields

lim k!k gr J (f 0 (k) ) 1 f 0 (k) (n + ) = h ((1 ) n) g (n) (66)
Then, (k gr ) does not exist, but

lim k!k gr (k) = 0 only if h ((1 ) n) = g (n). If h ((1 ) n) 6 = g (n)
, there is a unique other solution that solves J (f 0 (k) ) = 1 because J is U-shaped (see lemma 1). Let us denote k this solution, which is greater than k gr if and only if the slope of J in k gr is negative; formally one has k k gr , h (( 1) n) g (n) 0 (67)

Hence, whatever the relative position of k and k gr , it can be stated that for all k 2 0; k then J 2 (0; 1) if k > k gr and J > 1 if k < k gr . Conversely, for all k > k then J > 1 if k > k gr and J 2 (0; 1) if k < k gr . Therefore, conclude with (64) that (k) > 0 for all k 2 0; k .

Continuity of (k) for all k 2 0; k is guaranteed because, using (66), one has

lim k!k gr (k) = [f (k gr ) k gr f 0 (k gr )] [h ((1 ) n) g (n)] 1 (68)
Step 2. Since lim

k!0 [f (k) kf 0 (k)] = 0 and lim k!0 f 0 (k) = +1, observe that lim k!0 (k) can be 0. But since (k) > 0 for all k 2 0; k , a su¢ cient condition to have lim k!0 (k) > 0 is lim k!0 0 (k) < 0.
Function …rst derivative is

0 (k) = f 00 (k) (k) " k f (k) kf 0 (k) J 0 (f 0 (k) ) J (f 0 (k) ) 1 + 1 f 0 (k) ( + n) # (69) Then lim k!0 0 (k) < 0 if and only if lim k!0 " k f (k) kf 0 (k) J 0 (f 0 (k) ) J (f 0 (k) ) # < 0 (70)
Using l'Hôpital's rule and (62), yields condition [START_REF] Grandmont | Money and Value: a Reconsideration of Classical and Neoclassical Monetary Theories[END_REF].

(ii) Uniqueness. We give su¢ cient conditions that ensure 0 k < 1. We …rst study the case given by k > k gr then the one given by k < k gr (see condition (67)).

Case 1: k > k gr . In this case the equilibrium may be dynamically e¢ cient ( k < k gr or equivalently J > 1) or ine¢ cient ( k > k gr or J 2 (0; 1)). Computing (69) for k = k yields

0 k = kf 00 k J f 0 k J f 0 k 1 2 4 k f k kf 0 k J 0 f 0 k J f 0 k 3 5 (71)
Therefore 0 k < 0 if and only if

J 0 f 0 k J f 0 k k f k kf 0 k , k k gr (72)
Recall (lemma 1) that J 0 =J is increasing in f 0 k and therefore decreasing in k. Moreover, observe, using (68), that condition (72) is always veri…ed for k = k gr . Then, conclude that a su¢ cient condition for 0 k < 0 is

d dk k f (k) kf 0 (k) ! 0 , " (k) s (k) (73)
since condition (20) is assumed.

Case 2: k < k gr . In this case the equilibrium can only be dynamically e¢ cient ( k < k gr or J > 1) and J 0 =J is necessarily positive. It is easy to see that condi-tion (73) is not su¢ cient in this case because J 0 =J and k= [f (k) kf 0 (k)] may cross each other before k, opening then the possibility for multiple equilibria. Hence, use (71) to state that 0 k < 1 if and only if

J 0 f 0 k J f 0 k > k for k < k gr (74) with k k f k kf 0 k 2 4 1 1 J f 0 k 3 5 1 kf 00 k (75)
A su¢ cient condition for uniqueness is then 0 k > 0 because (using l'Hôpital's rule)

lim k!0 k = lim k!0 " k f (k) kf 0 (k) 1 kf 00 (k) # = 0 (76)
This implies that if there exists a ki such that 0 ki > 1, then all kj , such that kj > ki , will verify 0 kj > 1. Since there exists a …nite k and since (k) is continuous over 0; k , this is impossible. To obtain a nice condition implying 0 k > 0, rewrite (75) such that

(k) k f (k) kf 0 (k) ( 1 " 1 1 J (f 0 (k) ) # " (k) s (k) ) (77)
where " and s are de…ned in the core of the text. Then it is su¢ cient for (k) to increase in k that (73) be satis…ed and that

d dk s (k) " (k) ! 0 (78)
Hence, (73) and (78) are su¢ cient for uniqueness.

The following …gures give a graphical intuition of this proof: if (k) is continuous and positive for all k 2 0; k , it is su¢ cient that lim Its …rst derivative with respect of k is given by ))] J (f 0 (k) )

dc 2 (k) dk = f 0 (k) (n + ) 1 1=J (f 0 (k) ) + kf 00 (k) 1 1=J (f 0 (k) ) f 00 (k) J 0 (f 0 (k) ) [J (f 0 (k) )] 2 k [f 0 (k) (n + )] [1 1=J (f 0 (k) )] 2 (
where J, g and h are de…ned in lemma 1. At the equilibrium such that k = k; n one may, using equations ( 17) and [START_REF] Ghiglino | Dynamics in OG economies[END_REF], rewrite (82) such that

0 n k; n k = J f 0 k J f 0 k 1 " g (n) h n f 0 k + k ĉ # ( 83
)
where ĉ stands for the aggregate consumption. Now turn to the de…nition of the average age of capital holders using [START_REF] Lee | Age structure, intergenerational transfers and economic growth: an overview[END_REF].

Integrating by parts the numerator, using a (0) = p ( ) = 0; yields Then, since a = k and using factor prices equations, it yields

h (n r) = (f 0 (k) (n + )) a k c + g (n) f (k) kf 0 (k) c + k c (86)
Replace (86) in (83) and arrange, using (57) and recalling that g (n) = w , to obtain

0 n k; n = k [ a w ] (87)
replace in (81); this yields [START_REF] Malinvaud | Macroeconomic theory[END_REF].

PROOF. [Proof of proposition 5] We demonstrate it exists a n that solve 0 n k; n = 0 (88) with 0 n de…ned by (83). Let be such that f 0 k = + . Since there is no restrictions on n at the equilibrium, it is su¢ cient for the existence of n to show that: (i) which, using (67) and (73) and recalling that h 0 < 0, is negative and …nite.

(ii) Inequality n > is equivalent to k gr < k. Hence note with (83) that since

ĉ = f k (n + ) k (91)
we have to prove that

h n f 0 k g (n) k f k (n + ) k , k k gr (92)
Recall with (68), that the equality between the two terms of (92) is guaranteed when k = k gr . Then, observing that the left hand side is decreasing in k (because h 0 < 0) and that the right hand side is increasing in k, is su¢ cient to prove that (92) always holds.

(iii) Recall …rst (lemma 1) that

lim n! 1 g (n) = lim n! 1 h n f 0 k = (93)
Hence, with (83)

lim n! 1 0 n k; n = lim n! 1 k f 0 k (n + ) (94)
which is necessarily positive.

PROOF. [Proof of proposition 6] This is proven by the point (ii) of the proof of proposition 5.

PROOF. [Proof of proposition 7] As a preliminary, let us denote 0 n k; n; the …rst derivative of (34) with respect to n; the same kind of computations as those done in the proof of proposition 4 yield that To show that dn =d > 0, we demonstrate that @ @n 0 n k; n ; < 0 (97) and that @ @ 0 n k; n ; > 0 (98) Condition (97) is trivial since 0 n is continuous in n and since it is assumed there exists a unique n . If there are multiple solutions to (96), one should consider the one that maximizes k and the result would be the same.

0 n k; n; k = J f 0 k J f 0 k 1 " g (n ) h n f 0 k + k ĉ # (
To prove condition (98), derive (95) when n = n (i.e. when 0 n k; n ; = 0) to obtain

@ @ 0 n k; n ; = k J f 0 k J f 0 k 1 g 0 (n ) (99)
which is positive because g 0 < 0 (lemma 1) and J > 1 for all n < . This establishes the proof.

  It is moreover necessary to replace condition (4) by the No-Ponzi-Game condition: lim !+1

Proposition 3

 3 gives existence and uniqueness conditions for a non-trivial equilibrium. Technically, condition (20) implies lim k!0 (k) = +1 and conditions (21) ensure 0 k < 1. Let us now turn to the interpretation.

  Figure 1

k! 0 (

 0 k) > 0 to have at least one equilibrium. Moreover, if there are multiple equilibria, the one with the greater k must verify 0 k < 1. Examples of unique equilibrium and multiple equilibria are respectively presented in Figure (2-a) and (2-b).

Derivation of …gure 1

 1 Figure 2-a Figure 2-b

equivalent to 0 k < 1 . 0 n k; n 1 0

 101 79)Condition[START_REF] Grandmont | Money and Value: a Reconsideration of Classical and Neoclassical Monetary Theories[END_REF] ensures c 2 (0) = 0. Then, let, k be such that c 1 k = c 2 k ; simple manipulations of (79) show that Hence, if[START_REF] Judd | The law of large numbers with a continuum of i.i.d. random variables[END_REF] is veri…ed, c 1 (k) and c 2 (k) cross each other only once and …gure 1 follows.PROOF. [Proof of proposition 4]We have de…ned k such that k = k; n .This proof details the simple computing that leads to[START_REF] Malinvaud | Macroeconomic theory[END_REF]. Application of implicit functions theorem yieldsd k dn = k k; n(81)Deriving equation[START_REF] Ghiglino | Dynamics in OG economies[END_REF] with respect to n yields0 n (k; n) = 1 f 0 (k) (n + ) (k; n) (82) + f (k) kf 0 (k) f 0 (k) (n + ) [g (n) h (n (f 0 (k)

0 (

 0 j ) p ( ) exp ( n ) a ( ) d n R 0 p ( ) exp ( n ) a ( ) d + R 0 p ( ) exp ( n ) a 0 ( ) d n R 0 p ( ) exp ( n ) a ( ) d (84)Replace condition (2) and rearrange using (59), (60) and[START_REF] Buiter | Death, population growth and debt neutrality[END_REF] to obtain

  95) where ĉ = f k (n + ) k. Hence, we de…ne n such that 0 n k; n ; = 0 (96)

  0 n is continuous in n; (ii) 0 n k; n < 0 for all n > ; (iii) To show the continuity of 0 n in n, we simply have to prove that 0 n (k gr ; n)1 when k gr < k. Indeed, observe with (65) and (68) that both numerator and denominator of (83) equal zero, for k = k gr . Applying l'Hôpital's rule yields (k gr ) k gr f 0 (k gr )

			lim n! 1	0 n k; n > 0	(89)
	(i) lim k!k gr	0 n k; n =	k gr ) n) g (n) ) n) [g (n) h ((1 + 1 h 0 ((1 h ((1 8 > < > : f 00 (k gr ) d dk k gr f 9 ) n)] 2 > = > ;	(90)

See Cass[START_REF] Cass | Optimum growth in an aggregative model of capital accumulation[END_REF], Koopmans[START_REF] Koopmans | On the concept of optimal economic growth[END_REF].

Further developments have been made recentely by Burke[START_REF] Burke | Equilibrium for overlapping generations in continuous time[END_REF], Malinvaud[START_REF] Malinvaud | Macroeconomic theory[END_REF], Bommier and Lee[START_REF] Bommier | Overlapping generations models with realistic demography: statics and dynamics[END_REF] and Demichelis and Polemarchakis[START_REF] Demichelis | Frequency of trade and the determinacy of equilibrium paths: logarithmic economies of overlapping generations under certainty[END_REF].

These were …rst introduced in economic models byArthur and McNicoll [2] and Lee[START_REF] Lee | Age structure, intergenerational transfers and economic growth: an overview[END_REF].

Our problem is less general, however, since we are considering a speci…c instantanous utility function, namely a CRRA with 2 (0; 1]. Relaxing this assumption drastically complicates the proof of the existence of an aggregate equilibrium.

See the discussion in Judd[START_REF] Judd | The law of large numbers with a continuum of i.i.d. random variables[END_REF] and Feldman and Gilles[START_REF] Feldman | An expository note on individual risk without aggregate uncertainty[END_REF].

The left hand side of (14) is a positive, decreasing function of n, whose limits are respectively +1 and 0 when n goes to 1 and +1.

See the appendix for the details.

Proposition 2.4 page 83.

It is only at the limit (i.e., in a discrete time model, when the population growth rate goes to 1) that both ages would be equal.

See e.g. Seierstad and Sydsaeter[START_REF] Seierstad | Optimal control theory with economic applications[END_REF].

Acknowledgements

I wish to acknowledge receiving stimulating remarks and helpful suggestions from an associate editor and an anonymous referee of this review. This paper constitutes a modi…ed version of the third chapter of my Ph.D. dissertation that I have defended at the Université Paris-I Panthéon-Sorbonne. I am grateful to my supervisor K. Schubert as well as to A. d'Autume, A. Bommier, B. Decreuse, J-P. Drugeon and to seminar participants at Brown and Paris I Universities for their insightful comments. The usual disclaimer applies.


Stéphane Clain
email: clain@math.uminho.pt

Steven Diot
email: steven.diot@math.univ-toulouse.fr

Raphael Loubère
email: raphael.loubere@math.univ-toulouse.fr

Gaspar Machado

Rui Ralha
email: r_ralha@math.uminho.pt

Rui M S Pereira

Very high-order finite volume method for one-dimensional convection diffusion problems

Keywords: Finite Volume, very high-order, convection-diffusion, polynomial reconstruction

We propose a new finite volume method to provide very high-order accuracy for the convection diffusion problem. The main tool is a polynomial reconstruction based on the mean-value to provide the best order. Numerical examples are proposed to show the method efficiency.

Introduction

Efficient numerical schemes to solve convectiondiffusion problems are a constant challenge due to the wide range of problems which concern the coupling of the two major physical phenomena. Finite differences and finite elements are very popular to produce numerical approximations ( [START_REF] Chen | Finite element methods and their applications[END_REF][START_REF] Grossmann | Numerical treatment of partial differential equations[END_REF][START_REF] Tveito | Introduction to partial differential equations: a computational approach[END_REF]) and a lot of academic or commercial codes are based on such techniques.

In the last decade, the finite volume method appears to be an interesting alternative for several reasons, the simplicity (one information per cell), the built-in conservative property, the capacity to handle unstructured and non conformal meshes. Important developments have been realized ([4, 5, 6]) in this way but a serious drawback of the finite volume method is the large amount of numerical viscosity and the weak convergence rate (at most second-order convergence). The paper is devoted to a new class of finite volume schemes for convection-diffusion problem able to reach the sixth-order accuracy. We present the method for the simple one-dimensional case and restrict to the steady-state case for the sake of consistency. The main tool is an accurate local polynomial reconstruction based on the mean values approximations in place of the pointwise values approximations.

The rest of the paper is as follows. The second section recalls the classical finite volume scheme for convection diffusion problem (namely the Patankar method). Then, we introduce the polynomial reconstruction and the high-order schemes in section three. The last section is devoted to some numerical tests to show the scheme capacity to provide sixth-order accuracy.

Classical finite volume schemes

For the sake of simplicity, we consider the onedimensional convection-diffusion problem in domain Ω :=]0, 1[ with Dirichlet boundary conditions:

-(au ) + (vu) = f on Ω u(0) = u lf , u(1) = u rg ,
where we assume that a and v are regular functions on Ω with a(x) ≥ α > 0 for all x ∈ Ω. We denote by T h a mesh of Ω constituted of cells K i := [x i-1/2 , x i+1/2 ], i := 1, . . . , I, of centroid c i , where x 1/2 := 0 and x i+1/2 := x i-1/2 + h i , and set h := min I i=1 h i as the characteristic length of the mesh.

In the finite volume context, u i denotes an approximation of the mean value over cell K i , that is

u i ≈ 1 h i K i u dx.
We recall the classical scheme for the interior cells K i , i := 2, . . . , I -1:

-F d (u i , u i+1 ; x i+ 1 2 ) -F d (u i-1 , u i ; x i-1 2 ) + F c (u i , u i+1 ; x i+ 1 2 ) -F c (u i-1 , u i ; x i-1 2 ) = f i ,
where the diffusion and the convection fluxes write

F d (u i , u i+1 ; x i+ 1 2 ) := a(x i+ 1 2 ) 2(u i+1 -u i ) (h i + h i+1 ) , F c (u i , u i+1 ; x i+ 1 2 ) := [v(x i+ 1 2 )] + u i + [v(x i+ 1 2 )] -u i+1 , with [α] + := α+|α| 2
and [α] -:= α-|α| 2 , while f i denotes an approximation of the mean value of f over cell K i , that is f i ≈ 1 h i K i f dx (for the first cell and the last cell the expressions change in the classical way). We obtain the following linear system

A U = F + F D with U := (u 1 , . . . , u I ) t ∈ R I the unknown values, F := (f 1 , . . . , f I ) t ∈ R I the source term and
F D := 2a(0) h 1 + [v(0)] + u lf , 0, . . . , 0, 2a (1)
h I + [v(1)] -u rg t ∈ R I
the vector collecting the contributions deriving from the Dirichlet boundary conditions. Remark that the finite volume scheme is not equivalent to the finite element scheme or finite difference one for non-uniform meshes.

Very high-order schemes

Very high-order schemes are based on local polynomial reconstruction. For each cell K i , i := 1, . . . , I, we denote by u i (x) ∈ P d the polynomial reconstruction on K i , where d is the polynomial degree. The reconstruction we detail in the sequel is a linear operator P, that from U := (u 1 , . . . , u I ) t and Dirichlet conditions u lf , u rg provides u i (x) ∈ P d .

Based on the reconstruction, we then define the new numerical fluxes. For the interior cells we have

F d d (x i+ 1 2 ) := a(x i+ 1 2 ) u i (x i+ 1 2 ) + u i+1 (x i+ 1 2 ) 2 , F d c (x i+ 1 2 ) := [v(x i+ 1 2 )] + u i (x i+ 1 2 ) + [v(x i+ 1 2 )] -u i+1 (x i+ 1 2 ),
while for the first cell and the last cell we consider, for the diffusive flux the expressions

F d d (0) := a(0) u 1 (0), F d d (1) := a(1) u I (1).
At last, we define the affine operator

U → G d (U ; u lf , u rg ) from R I into R I with G d i (U ; u lf , u rg ) := -F d d (u i , u i+1 ; x i+ 1 2 ) -F d d (u i-1 , u i ; x i-1 2 ) + F d c (u i , u i+1 ; x i+ 1 2 ) -F d c (u i-1 , u i ; x i-1 2 ) -f i
To provide the numerical solution, one has to find the solution U of the linear problem

G d (U ; u lf , u rg ) = 0 I := (0, . . . , 0) t . (1)
We now detail the polynomial reconstruction which is the crucial point of the numerical technique. We shall consider two different situations whether we take the boundary cells (K 1 and K I ) into account or the interior cells (K i , i := 2, . . . , I -1).

The general case concerns the inner cells. To this end, for each cell K i we associate a index subset ν(i) such that for a given polynomial degree d, we choose the nearest d + 2 cells to K i . We then consider the polynomial function of the form

u i (x) := u i + d α=1 R i α (x -c i ) α - 1 h i K i (x -c i ) α dx
and we seek the coefficients R i α , α := 1, . . . , d, which minimize the functional

E(R i α ) := j∈ν(i) 1 h j K j u i dx -u j 2 .
The important point is that we minimize the functional with respect to the mean values and not identify (as usually done in finite volume context) the mean values with pointwise values at the centroid.

For the first cell, we slightly modify the minimizing functional. We associate the index subset ν(1) := {2, . . . , d -1} and augment the functional with the Dirichlet condition

E(R 1 α ) := ( u 1 (0) -u lf ) 2 + j∈ν(1) 1 h j K j u 1 dx -u j 2 .
The reconstruction for the last cell is analogous to the first cell.

A simple way to solve the affine problem (1) consists in evaluating the associated matrix A d and the right-hand side F d and then to solve the linear problem A d U = F d using standard methods such as the LU method. To this end, we first compute the righthand side setting F d := G d (0 I ; u lf , u rg ), then we evaluate each column A d j of the matrix A d with

A d j := G d (e j ; u lf , u rg ) -F d ,
where e j is the canonical vector with the j-component equal to 1.

It should be stressed that we construct the associated full matrix A d for the sake of simplicity. In matter fact, since we only focus on the numerical scheme itself, that construction provides a simple method to test the convergence rate.

Numerical tests

To compare the numerical approximation with the exact solution, we introduce three kind of errors, namely the consistency error, the L ∞ error of the solution and the L ∞ error of the derivative (we intentionally use the L ∞ -norm since we deal with regular functions). More precisely, consistency error is given by

E C (U ) := I max i=1 G d i (U ; u lf , u rg ) , (2)
where

U := (u 1 , . . . , u I ), u i := 1 h i K i u dx,
are the exact mean values of the solution of the continuous problem. The L ∞ error between the mean values is given by

E 0 := I max i=1 |u i -u i |,
while we define the L ∞ error for the derivative with

E 1 := I max i=1 u i (x i-1 2 ) -u (x i-1 2 ) , u i (x i+ 1 2 ) -u (x i+ 1 2
) .

We consider a uniform mesh of I cells and test three types of polynomial reconstructions, namely the P 1 , P 3 , and P 5 .

The first case concerns the exponential function with constant diffusion and velocity, i.e., a := 1 and v := 1 leading to a Peclet number of Pe=1. We state that the solution is exp(x) and deduce that f = 0, u lf = 1, and u rg = e. 10 9.7e-04 -6.0e-05 -4.0e-04 -20 1.3e-04 2.9 4.7e-06 3.7 5.5e-05 2.9 40 1.7e-05 2.9 3.3e-07 3.9 7.2e-06 2.9 80 2.2e-06 3 2.2e-08 3.9 9.3e-07 3 P 5 10 4.4e-06 -1.0e-07 -1.4e-06 -20 1.6e-07 4.8 2.3e-09 5.5 5.2e-08 4.8 40 5.3e-09 4.9 4.1e-11 5.8 1.7e-09 4.9 80 1.7e-10 5 6.8e-13 5.9 5.6e-11 4.9

Figure 1: Example 1 -convergence curves of E 0 for the exponential function using a uniform mesh with Pe=1.

Figure 2: Example 1 -convergence curves of E 1 for the exponential function using a uniform mesh with Pe=1.

Table 1 provides the consistency error and the numerical errors both for the function and its derivative for the L ∞ norm while we plot in Figures 1 and2 the convergence curves for the solution and its derivative. We obtain an effective d + 1-order of convergence rate for the functions and d-order convergence rate for the solution derivative when employing the reconstruction with polynomial functions in P d . Consistency error also enjoys the convergence property since the scheme provides the optimal d-order rate (remember that the consistency error (2) contains derivatives).

The second case concerns the sine function with constant diffusion and velocity a := 1 and v := 100 such that Pe=100. The goal is to handle problems with a large Peclet number providing a convectiondominated problem. We state that the solution is sin( πx 2 ) and deduce that f (x) = π For a large Peclet number, the numerical schemes manage to provide the optimal order convergence, with the exception of the P 1 case, as can be seen in Table 2 and Figures 3 and4. For the P 1 polynomial reconstruction, large error are derived from the conflicting outflow boundary condition since the fluid flow out at x = 1. A pure convection problem does

Conclusions and further work

We have presented a new finite volume method for one-dimensional convection-diffusion problem which provides very high-order accuracy. Numerical simulations have been carried out to show the capacity of the method to provide a sixth-order scheme. Extensions for two-dimensional domains and the consideration of both Dirichlet and Neumann boundary conditions are under investigation. Another difficulty concerns the solution stability when dealing with rough data. A strategy based on the MOOD method ( [START_REF] Clain | A high-order polynomial finite volume method for hyperbolic system of conservation laws with Multidimensional Optimal Order Detection (MOOD)[END_REF][START_REF] Clain | Multi-dimensional Optimal Order Detection (MOOD) -A very high-order Finite Volume Scheme for conservation laws on unstructured meshes[END_REF]) is also under investigation.

2 2 sin πx 2 + 50π cos πx 2 ,

 22 u lf = 0, and u rg = 1.

5 Figure 3 :

 53 Figure 3: Example 2 -convergence curves of E 0 for the sine function using a uniform mesh with Pe=100.

Figure 4 :

 4 Figure 4: Example 2 -convergence curves of E 1 for the sine function using a uniform mesh with Pe=100.

Table 1 :

 1 Example 1 -errors and rates of convergence for the exponential function using a uniform mesh with Pe=1.

		I	E C		E 0		E 1
			err	ord	err	ord	err	ord
		10 7.8e-02 -8.3e-03 -7.4e-02 -
	low	20 4.2e-02 0.9 4.6e-03 0.8 3.8e-02 1 40 2.2e-02 0.9 2.5e-03 0.9 1.9e-02 1
		80 1.1e-02 1 1.3e-03 1 9.8e-03 1
		10 1.8e-01 -1.4e-01 -3.2e-01 -
	P 1	20 9.1e-02 1 4.9e-02 1.5 1.5e-01 1.1 40 4.6e-02 1 1.7e-02 1.5 7.3e-02 1.1
		80 2.3e-02 1 5.8e-03 1.5 3.5e-02 1
	P 3

Table 2 :

 2 Example 2 -errors and rates of convergence for the sine function using a uniform mesh with Pe=100.

		I	E C		E 0		E 1
			err	ord	err	ord	err	ord
		10 7.8e+00 -6.5e-02 -1.3e+00 -
	low	20 3.9e+00 1 2.8e-02 1.2 1.1e+00 0.2 40 2.0e+00 1 1.1e-02 1.4 8.7e-01 0.4
		80 9.8e-01 1 5.2e-03 1.1 6.0e-01 0.5
		10 2.9e-01 -3.9e-03 -2.0e-01 -
	P 1	20 2.

9e-02 3.3 5.2e-04 2.9 1.1e-01 0.9 40 1.4e-02 1.1 1.3e-04 2 5.6e-02 0.9 80 1.4e-02 0 1.8e-04 0.5 3.0e-02 0.9 P 3 10 5.1e-03 -5.3e-05 -1.0e-03 -20 2.9e-04 4.1 2.9e-06 4.2 1.3e-04 2.9 40 2.1e-05 3.8 1.6e-07 4.2 1.8e-05 2.9 80 3.1e-06 2.7 1.1e-08 3.9 2.4e-06 2.9

Acknowledgements: This research was financed by FEDER Funds through Programa Operacional Factores de Competitividade -COMPETE and by Portuguese Funds through FCT -Fundação para a Ciência e a Tecnologia, within the Project PEst-C/MAT/UI0013/2011.


' Albis

Emmanuelle Augeraud-Véron

Hippolyte D'albis
email: hippolyte.d-albis@univ-tlse1.fr

in an OLG model with a continuum of …nitely-lived individuals

Keywords: Overlapping-generations Models, Mixed-type Functional Di¤erential Equations, Endogenous Fluctuations, Hopf Bifurcation. JEL Classi…cation Numbers: C6, D9

Balanced cycles

Introduction

This paper investigates the long-run ‡uctuations that may emerge in stationary OLG economies. To this end, we follow Demichelis and Polemarchakis [START_REF] Demichelis | Frequency of trade and the determinacy of equilibrium paths: logarithmic economies of overlapping generations under certainty[END_REF] who study a continuous-time model with a continuum of …nitely-lived individuals 1 . The intertemporal equilibrium is shown to be the solution of a functional di¤erential equation of mixed-type (MFDE). By extending this model to isoelastic preferences, the MFDE is nonlinear. The proof of the existence of long-run ‡uctuations then relies on Rustichini [START_REF] Rustichini | Hopf bifurcation for functional di¤erential equations of the mixed type[END_REF] and Benhabib and Rustichini [START_REF] Benhabib | Vintage capital, investment and growth[END_REF]: it uses the linearized MFDE that characterizes the local motion of the economy and looks for solutions which have Hopf bifurcation values. Assuming zero discounting and a speci…c endowment distribution, we show that there exist some sets of parameters such that a barter economy exhibits a cycle on the neighborhood of a steady state. An elasticity of intertemporal substitution lower than one and zero endowments at beginning and end of the individuals'life-span are su¢ cient conditions to obtain this result. This paper consequently shows that an increase in the frequency of trade during the individual' life-span does not eliminate the possibility of endogenous ‡uctuations in OLG economies 2 .

The sketch of the paper is as follows. Section 2 presents an overlappinggenerations model with continuous trading and …nitely-lived individuals. Sec-tion 3 characterizes the intertemporal equilibrium of a barter economy and analyses the existence and uniqueness properties of the steady state. Section 4 studies the spectral decomposition of the linearized dynamics in the neighborhood of a steady state and gives su¢ cient conditions for the existence of an Hopf bifurcation.

The model

u (s) = Z s+1 s c (s; t) 1 1 1 1 dt (1)
where > 0 stands for the elasticity of intertemporal substitution. During a lifetime, an age-dependent endowment is received; it is denoted w (t s)

and satis…es:

w (t s) = w (t) if t 2 [s + ; s + ] 0 otherwise (2)
with 0 < 1. Individuals have access to a competitive asset market that yields the interest rate r (t). Let a (s; t) denotes the real wealth of an individual who born at time s as of time t. The instantaneous budget constraint is therefore:

@a (s; t) @t = r (t) a (s; t) + w (t s) c (s; t) (3)
Individuals are born with no …nancial assets and cannot die indebted. Therefore, initial and terminal conditions write:

a (s; s) = 0 (4)
a (s; s + 1) 0 (5)
It is assumed that a (s; t) and c (s; t) are C 1 R 2 + and that r (t) and w (t s)

are continuous for all t 2 [s; s + 1]. The individual program is to maximize

(1) subject to (3), ( 4) and ( 5).

Lemma 1

The optimal consumption pro…le satis…es:

c (s; t) = R s+ s+ w (z) e R z s r(u)du dz R s+1 s e (1 )
R z s r(u)du dz e R t s r(u)du (6)
Proof: The …rst order conditions are:

@c (s; t) @t = r (t) c (s; t) (7)
and a (s; s + 1) = 0. Integrating forward condition (3), and using the second optimal condition yields:

Z s+1 s c (s; z) e R z s r(u)du dz = Z s+ s+ w (z) e R z s r(u)du dz (8)
Replacing [START_REF] Kitagawa | Endogenous growth cycles in an overlapping generations model with investment gestation lags[END_REF] in (8) gives c (s; s) and consequently [START_REF] Ghiglino | Dynamics in OG economies[END_REF].

The demographic structure is in overlapping generations. Each individual belongs to a cohort whose size is normalized to 1. There is no population growth and, at each point of time, a new cohort enters the economy since the oldest one leaves it. Hence, the aggregate counterpart, x (t), of any individual variable, x (s; t) ; is obtained by integrating over the birth date, such that:

x (t) = Z t t 1
x (s; t) ds

Assume there exists a single non storable good and that the aggregate endowment equals the size of the population; then: R t t 1 w (t s) ds = 1. Replacing the endowment distribution rule given by (2) yields: w (t) = 1= ( ). Using ( 6), the aggregate consumption, denoted c (t), hence satis…es:

c (t) = 1 ( ) Z t t 1 R s+ s+ e R z s r(u)du dz R s+1 s e (1 )
R z s r(u)du dz e R t s r(u)du ds (10)
There is no money in this economy and assets are constituted by consumption loans. The aggregate wealth is denoted a (t); using (3), (4) and the optimal condition: a (s; s + 1) = 0, its dynamics writes:

da (t) dt = r (t) a (t) + 1 c (t) (11)
3 Equilibrium and steady states De…nition 1 An equilibrium with perfect foresight is a function

F (t) = (c (t) ; a (t) ; r (t)), F : R + ! R 2 + R, C 1 (R + ) such that (i) individuals max-
imize their utility subject to the budget constraint, (ii) the aggregate consumption equals the aggregate endowment (i.e. c (t) = 1), and (iii) the aggregate asset is zero (i.e. a (t) = 0).

Using (3), it can be inferred from de…nition 1, that an equilibrium reduces to a function r (t).

De…nition 2 A steady state equilibrium is a triplet (c; a; r) that satis…es:

(i) c = (r) and a = (c 1) =r, (ii) c = 1 and a = 0; where function : R ! R ++ is such that:

(r) = R 1 0 e rz dz ( ) R e rz dz R 1 0 e (1 )rz dz (12)
Property 1 A steady state interest rate is a r that satis…es (r) = 1 if r 6 = 0 and 0 (r) = 0 if r = 0.

Proof: For r 6 = 0, the property is an immediate implication of de…nition 2;

for r = 0; just observe that (0) = 1 and use l'Hôpital's Rule.

Lemma 2 r = 0 is a steady state interest rate if and only if + = 1.

Proof: Replace r = 0 in 0 (r) and use property 1 to conclude.

Corollary 1 The individual consumption is age-independant if and only if

+ = 1.
Lemma 3 There exist ( ; ; ) such that there is no steady-state equilibrium.

Proof: It can be easily shown that there is no solution r 6 = 0 such that (r) = 1 if = 0 and 2 (0; 1 ] or if = 1 and 2 [ ; 1).

Lemma 4 There exist ( ; ; ) such that there are multiple steady-state equilibria.

Proof: Suppose = 1 and recall with lemma 2 that r = 0 is a steady state; it can be easily shown that there exist at least two other solutions )).

r 6 = 0 such that (r) = 1 if 2 ( ; 2 (1

Endogenous cycles

The equilibrium satis…es the following non linear dynamics:

1 ( ) Z t t 1 R s+ s+ e R z s r(u)du dz R s+1 s e (1 )
R z s r(u)du e R t s r(u)du ds = 1 (13)
In this section, it is the local dynamics around steady-state r which is studied; it is the one of x (t) de…ned such that r(t) = r + x(t).

Property 2 The characteristic function of x (t) is denoted Q( ) and satis…es:

Q( ) = R e ( r )s dz R e r z dz + (1 ) R 1 0 e ( (1 )r )z dz R 1 0 e (1 )r z dz + R 1 0 e r z dz R 1 0 e ( r )z dz (14)
Proof: Replace r(t) = r + x(t) in (13). Then do a Taylor expansion in the neighborhood of = 0 and rearrange using (r) = 1. De…ne X(t) = R t 0 x (u) du. It yields:

X(t) = R t t 1 e r (t s) R s+ s+ e r (z s) X(z)dz ( ) (1 ) R s+1 s e (1 )r (z s) X(z)dz R 1 0 e r z dz ds R 1 0 e (1 )r z dz (15)
Finally, Q( ) is obtained by the following change of variable: x(t) = e t and rearranging using (r) = 1.

Lemma 5

The characteristic function Q( ) has an in…nity of complex roots with negative real parts and an in…nity of complex roots with positive real parts.

Proof: Roots of Q( ) are asymptotic to those of the following equations 3 :

P 1 ( ) = 2 Z 1 0 e r z dz + (1 ) e (1 )r R 1 0 e (1 )r z dz ! (16) P 2 ( ) = 2 Z 1 0 e r z dz + (1 ) e + r R 1 0 e (1 )r z dz ! (17)
Asymptotically, roots of P 1 ( ) have a positive real part while those of P 2 ( ) have a negative real part.

Corollary 2

The dynamics is generically characterized by oscillations that decrease in magnitude and eventually disappear.

Consider now the case = 1 and focus on the neighborhood of the steady state r = 0. The characteristic function (14) rewrites:

Q( ) = R 1 e z dz ( 1 2 )
+ ( 1)

Z 1 0 e z dz + 1 R 1 0 e z dz (18)
Lemma 6 There exist ( ; ; ) such that there are pure imaginary roots which are Hopf bifurcation values.

Proof: Let = p+iq. The proof proceeds in two steps: (i) it supposes = iq and proves that for < 2 (1 ), there exists a q > 0 such that Q(iq) = 0;

it hence de…nes ( 0 ; 0 ( 0 )) the pair of parameters for which this root does exist; (ii) it uses as a bifurcation parameter and shows that there exists a neighborhood of 0 such that d Re (Q ( )) =d is not equal to zero.

(i) Replace = iq in (18) and de…ne Q (q) such that:

Q (q) = cos ( q) cos ((1 ) q) 1 2 + (1 ) [cos (q) 1] q 2 2 (19) 3 See Bellman and Cooke (1963) p. 410.

The pioneer OLG model of this kind is due to Cass and Yaari[START_REF] Cass | Individual saving, aggregate capital accumulation, and e¢ cient growth[END_REF].

Note that this issue is usually debated in discrete time environment: see notably Aiyagari[START_REF] Aiyagari | Can there be short-period deterministic cycles when people are long lived?[END_REF] and Ghiglino and Tvede[START_REF] Ghiglino | Dynamics in OG economies[END_REF] in a framework with many generations and[START_REF] Michel | Optimal growth and cycles in overlapping generations models[END_REF],[START_REF] Kitagawa | Endogenous growth cycles in an overlapping generations model with investment gestation lags[END_REF] and[START_REF] Rochon | Debt, liquidity and dynamics[END_REF] in models with two generations.

Ricci for helpful comments and suggestions. The usual disclaimer applies

A Taylor expansion in the neighborhood of q = 0 yields:

Moreover, lim q!+1 Q (q) = 1. Therefore, there exists a q > 0 such that

(ii) From ( 18), it yields:

The roots of this latter function are those of sin (q) and cos (q) = 1 q 2 =2.

They are the f2 + k ; k 2 Zg [ fq 1 ; q 2 ; q 3 ; q 4 g, where q 1 ; q 2 ; q 3 and q 4 are the 4 roots of the following polynomial:

Since elements of f2 + k ; k 2 Zg [ fq 1 ; q 2 ; q 3 ; q 4 g are independent with respect to a 0 , there exists a 0 , such that a 0 = 2 f2 +k ; k 2 Zg[fq 1 ; q 2 ; q 3 ; q 4 g.