Millian Efficiency with Endogenous Fertility by J. Ignacio Conde-Ruiz ** Eduardo L. Giménez ** Mikel Pérez-Nievas DOCUMENTO DE TRABAJO 2004-13

October 2004

* FEDEA (Fundación de Estudios de Economía Aplicada); conde - ruiz@fedea.es ** Universidade de Vigo; egimenez@uvigo.es *** Universidade de Santiago de Compostela; aepmikel@usc.es

Millian Efficiency with Endogenous Fertility∗

J. Ignacio Conde-Ruiz†, Eduardo L. Giménez‡and Mikel Pérez-Nievas§

october 2004

Abstract

This paper studies an extension of the notion of Pareto eficiency, referred to as Millian eficiency, to evaluate the performance of symmetric allocations in an overlapping generations setting with endogenous fertility. The criterium of Pareto dominance underlying the notion of Millian eficiency is based exclusively on preferences of those agents who are actually born, and allows only for welfare comparisons of symmetric allocations (i.e, allocations in which all living individuals of the same generation take the same decisions). The main contributions of the paper are the following. First, we provide necessary (static) and suficient (dynamic) conditions to determine whether an allocation is Millian eficient or not, and we show that the suficient conditions for dynamic eficiency ofered by Cass (1972) and Balasko and Shell (1980) cannot be straightforward applied when fertility is endogenous. Second, we extend the two Fundamental Theorems of Welfare Economics to a framework with endogenous population by characterizing Millian eficient allocations as the equilibria of a decentralized price mechanism. Finally, we discuss alternative extensions of the Pareto criterium that strengthen the Millian notion.

Keywords: Endogenous fertility, Pareto optimality, Dynamic eficiency. JEL: D61, D91, H21, J13

^∗Corresponding author: Mikel Pérez-Nievas, Departamento de Fundamentos da Análise Económica, Facultade de C.C. Económicas, Universidade de Santiago de Compostela, E15782 Santiago de Compostela SPAIN, e-mail: <aepmikel@usc.es>.

We are grateful for the useful insights of Michele Boldrin, Tim Kehoe and Victor Ríos-Rull. We also acknoledge comments by Fernando Del Río, Antonio Molina, José L. Moraga, Juan Pintos, Herakles Polemarchakis, Manuel S. Santos, the participants to the VI and VIII Workshop on Dynamic Macroeconomics, Soutomaior 2001 and 2003, to the 11th European Workshop on General Equilibrium Theory, Athens 2002, and to the 59th Econometric Society European Meeting, Madrid 2004. Financial support from the Spanish Minister of Science and Technology projects SEC-2002-03421 and SEC2003-08988 is acknowledged by the authors, and SEC-2002-04318-CO2-01 by the second author.

^†FEDEA (Fundación de Estudios de Economía Aplicada); conde − ruiz@fedea.es

^‡Universidade de Vigo; egimenez@uvigo.es

^§Universidade de Santiago de Compostela; aepmikel@usc.es

1 Introduction

Fertility decisions are intimate individual (or household) decisions. Despite economists like to consider this decision as rational and optimal for individuals, it may result in resource misallocations at the aggregate level, an issue that worldwide deeply concerns governments and international institutions. On the one hand, high fertility rates are considered as a breakdown to development, and Third World governments are advised for reducing them with active policies aimed to increase the wealth of their nations. On the other hand, as Western population becomes older and older, the financial foundations of PAYGO state-pension systems are beginning to crumble, and policymakers have started to think on implementing policies that increase fertility rates. Furthermore, paying money for kids is becoming a very popular political proposal all around Europe, and policies consisting of linking the pension benefit to the number of descendants have been taken into account (e.g., Germany).

Despite these concerns, we are not furnished with enough theoretical grounds to determine what the optimal size of the population is and, consequently, to claim that individual fertility choices lead to a too low or a too high population size. The main reason for this is that the standard Pareto dominance criterium underlying the notion of Pareto optimality exclusively allows one to rank allocations for which the set of agents is fixed, and therefore any two allocations associated with diferent fertility choices cannot be compared.

This paper pretends to fill this gap by studying a notion of eficiency, which we refer to as Millian eficiency, applicable to evaluate equilibrium allocations in an overlapping generations framework with endogenous fertility and capital accumulation. We focus in a framework in which: (i) fertility choices are selected from a continuum; (ii) all living agents of the same generation have the same preferences on consumption bundles, represented by a well behaved utility function; and (iii) children are a costly consumption good, and parents derive utility from the number of children they bear but not from the utility of their descendants.¹ In this setting, the notion of Millian eficiency results from combining

a) an extension of the Pareto dominance criterium, proposed also in a recent paper by Golosov, Jones and Tertilt (2004) and referred to as the dominance criterium, according to which an allocation is preferred to another if it is weakly preferred by all agents who are alive in the two allocations, and strictly preferred by some of these agents; and,

b) a constraint on the set of allocations that can be compared using the dominance criterium, which is restricted to be formed by all feasible allocations in which i) every two living agents of the same generation are treated equally and therefore obtain the same consumption bundles; and ii) the population size of each generation is strictly positive.

To be more precise, a Millian eficient allocation is a symmetric allocation with positive fertility rates for every period that is not dominated by any other symmetric allocation that also yields positive fertility rates. In the symmetric setting studied in the paper, the notion of Millian eficiency reduces to a simple criterium: an allocation is Millian eficient if there does not exist any other allocation that makes every living agent of every generation better of without making any agent of any generation worse of. Although other names (such as, for example, constrained eficiency) might be more informative of the normative principles underlying this notion of eficiency, we use the term Millian eficiency because it generalizes a notion of optimality, referred to as Millian optimality, which has been frequently used in the literature.²

¹This type of framework has been studied, among others, by Eckstein and Wolpin (1985), Nerlove, Razin and Sadka (1985), Eckstein, Stern and Wolpin (1988), Bental (1989), Michel and Pestieau (1993), Cigno (1992, 2000), Raut (1995), and Groezen, Leers and Medjam (2003). For alternative representation of fertility choices, see Barro and Becker (1989).

Once we adopt an extension of the notion of Pareto eficiency, exploring its properties in the framework studied in the paper involves an important dificulty: as in other overlappinggeneration economies, the double infinity of traders and dates makes it dificult to provide a complete characterization of eficient allocations (see Shell, 1971). As Balasko and Shell (1980) made clear, determining a set of conditions that are necessary for achieving eficiency is fairly simple: every eficient allocation must also be statically eficient, i.e., it cannot be improved upon by a reallocation of resources of a finite number of generations, and therefore every eficient allocation must be a solution to a sequence of welfare optimization problems in which the feasible set is bounded.

However, providing a set of suficient conditions guaranteeing that a statically eficient allocation is in fact eficient (or dynamically eficient) is subtler, since a statically eficient allocation might be improved upon a reallocation of resources involving an infinite sequence of intergenerational transfers. Some authors have aforded necessary and suficient conditions for Pareto eficiency in diferent overlapping generation settings, but all of them consider fertility as exogenous.³ As we show in the paper, considering fertility as an endogenous variable introduces non-convexities on the sequence of inequalities characterizing the set of feasible allocations, because some aggregate variables are the product of two endogenous decisions. Due to these non-convexities, the suficient conditions ofered by Cass (1972) and Balasko and Shell (1980) cannot be applied. In view of this, we provide an extension of Balasko and Shell’s suficient condition to non-convex settings.

Next, endowed with the tools necessary to determine what fertility choices might be considered as eficient, we explore under what conditions individual decisions lead to eficient choices. To do this, we adapt the Fundamental Theorems of Welfare Economics to a setting with endogenous fertility by characterizing every (statically) Millian eficient allocation as the equilibrium of a decentralized sequential price mechanism. Analogously to the case of economies with exogenous population, every Millian eficient allocation can be decentralized by initially selecting an appropriate sequence of intergenerational transfers, and by allowing then the agents to determine their consumption and investment decisions at competitive markets. Diferently from the standard exogenous population case, in which non-distorting intergenerational transfers must be lump-sum for all agents, an incentive scheme that links intergenerational transfers with fertility decisions is needed. More precisely, for every system of intergenerational transfers that achieves Millian eficiency, every middle-aged adult has to pay a lump-sum tax (or, in some cases, receive a lump-sum subsidy), while every old adult has to receive a subsidy (or pay a tax) which depends linearly of the number of children she decided to have. As a particular case, we also show that the allocation corresponding to a decentralized equilibrium with no intergenerational transfers (for which there is no need to subsidize or tax children) is (statically) Millian eficient. In contrast with other environments with incomplete markets, this particular case shows that the absence of a market (in this case, a market where ofspring may bargain with their parents the right to be born) does not yield any eficiency loss, at least if one is concerned with Millian eficiency.

To conclude the paper, we provide alternative extensions of the notion of Pareto eficiency to the environment studied throughout the paper, and we explore whether or not Millian eficient allocations are also eficient under these alternative criteria. We show that this is actually the case for an alternative extension of the Pareto dominance criterium, referred to as u dominance criterium (constructed from a previous assumption on the utility level u obtained by non-born agents), provided we keep the restriction imposing that only symmetric allocations can be ranked using that criterium. However, dropping out the symmetry restriction on the set of allocations that can be compared using either the dominance or the u dominance criterium yields striking results: a Millian eficient allocation cannot be eficient, and the set of symmetric allocations that can also be u eficient reduces to those in which all living agents obtain the utility level u. These findings have an important implication: despite circumscribing to symmetric allocations involves some welfare losses from the point of view of both the dominance and the u dominance criteria, dropping out this restriction might induce an order on the set of allocations with no max imal elements; that is, the set of eficient and u eficient allocations might be empty. All these results suggest that any extension of the Pareto criterium applicable to an environment with en dogenous fertility like the one studied in the paper should incorporate symmetry considerations.

²See Nerlove, Razin and Sadka (1982), Cigno (1992,2003) or Groezen, Leers and Medjam (2003).

³Phelps (1965) and Koopmans (1963) in a growth model, and Diamond (1965) in a productive overlappinggeneration model found suficient conditions for eficiency. The first complete characterization was provided by Cass (1972) in the context of a simple physical capital growth model and, later, Balasko and Shell (1980) in overlapping generations exchange economy. Other relevant extensions are those obtained by Galor and Ryder (1991), Chattopadhyway and Gottardi (1999) or Molina and Pintos (2003).

In the literature of overlapping generation economies with endogenous fertility, two diferent approaches to provide normative principles can be distinguished: a first approach identifies socially optimal allocations with steady state optimal allocations (also referred to as golden rule allocations), that is, allocations that maximize the utility obtained by a representative consumer among those feasible stationary allocations;⁴ while a second approach identifies optimal allocations with those maximizing a certain class of social welfare maximization problems, referred to as Millian or Benthamite depending on whether or not the welfare weight given to a generation in the social welfare function depends on the size of that generation.⁵ Neither one of these two approaches takes explicitly into account the problem of dynamic eficiency, nor the fact that the standard Pareto criterium is not straightforward applicable to environments in which the set of agents is endogenous.⁶

A remarkable exception within the literature of endogenous fertility is the recent paper by Golosov, Jones and Tertilt (2004). They consider a general overlapping generations economy in which fertility decisions are discrete, and assume that all potential agents –included those that will never be born– have well defined preferences. In this context, they analyze two extensions of the Pareto dominance criterium (referred to as dominance and dominance) that are closely related to our notions of dominance and u dominance. However, their assumption of a discrete set of potential agents brings with it considerable dificulties if one is concerned with identifying eficient allocations in overlapping generations settings with non-altruistic agents, as the one studied in this paper. In fact, we will show that in this type of framework, the notions of optimality arising from their notions of dominance and dominance involve some dificulties. Without further restrictions on the set of allocations that can be compared using these dominance criteria, their notion of dominance might induce a non transitive relation on the set of feasible allocations, and their notion of dominance might induce an order for which the set of maximal

⁴See e.g., Samuelson (1975, 1976), Deardof (1976), Eckstein and Wolpin (1985), Bental (1989) or Michel and Pestieau (1993).

⁵See e.g., Nerlove, Razin and Sadka (1982, 1985), Cigno (1992), Groezen, Leers and Medjam (2003) or Razin and Sadka (1995, Ch.5) for a survey.

⁶Raut (1992), and more recently, Michel and Wigniolle (2003) have also proposed a notion of eficiency (which they refer to as “Pareto optimality”) that coincides with our notion of Millian eficiency. However, their notion of Pareto optimality is not explicitly deduced from an extension of the Pareto criterium, as it is in this paper. Moreover, their treatment of the dynamic eficiency problem is substantially less general, since they restrict the analysis to stationary allocations (Raut), or to environments with CES utility and production functions (Michel and Wigniolle).

elements is empty.

The paper is organized as follows. In section 2, we introduce the model. Next, in section 3, we present the notion of Millian eficiency mentioned above and provide necessary and suficient conditions to determine whether an allocation is eficient in this sense. In section 4 we characterize Millian eficient allocations as the equilibria of a decentralized sequential price mechanism. In section 5, we provide alternative notions of eficiency, and explore whether Millian eficient allocations are eficient under these alternative criteria. Finally, section 6 presents the main conclusions of the paper and discusses possible extensions.

2 The Model: Assumptions and Definitions

Consider an overlapping generations economy with three generations of consumers (referred to as old adults, middle-aged adults and children) coexisting at each period .. Each generation is formed by a set of identical agents who live for three periods of time and whose size is determined endogenously. To be more precise, at each period t there exist old adults (who were born at date middle-aged adults (born at date and children (born at t). For each write

\[n _ {t} = \left\{ \begin{array}{c l} \frac {N _ {t}}{N _ {t - 1}}, & \text { if } N _ {t - 1} > 0 \\ 0, & \text { otherwise. } \end{array} \right.\]

That is, for each an average extended family formed by an old adult born at and their descendants living at t has members in their middle age and children. The number of old adults at is normalized to one, and the number of middle-aged adults at is given by the initial condition

Resources available can be described as follows. Middle-aged adults are endowed with one unit of time to work, which is supplied inelastically. At each period a perishable consumption good is produced using labor and physical capital invested in previous period t 1 as inputs,⁷ that is,

\[Y _ {t} = F _ {t} (K _ {t}, N _ {t - 1}),\]

where is total output, and is a concave constant return to scale production function. Physical capital is fully depreciated in the production process,⁸ and the stock of capital at period is given by the initial condition

For each , write and for the average levels of output and capital per old adult (or per extended family), that is,

\[y _ {t} ^ {o} = \left\{ \begin{array}{c l} Y _ {t} / N _ {t - 2}, & \text {if} N _ {t - 2} > 0, \\ 0, & \text {otherwise}; \end{array} \right.\]

and

\[k _ {t} ^ {o} = \left\{ \begin{array}{c l} K _ {t} / N _ {t - 2}, & \text { if } N _ {t - 2} > 0, \\ 0, & \text { otherwise }. \end{array} \right.\]

With this notation, output per family can be written as a function of capital per family and workers per family, that is

\[y _ {t} ^ {o} = F _ {t} (k _ {t} ^ {o}, n _ {t - 1}).\]

F^∗_t (Kt, Nt₋₁) ≡ Ft(Kt, Nt₋₁) + (1 − δ)Kt

⁷See Conde-Ruiz, Giménez and Pérez-Nievas (2004) for the case with human capital.

⁸This assumption is without loss of generality, and extending the model to allow for a constant depreciation rate δ simply requires redefining the production function as .

The aggregate output of the homogeneous good is used to finance aggregate investments in physical capital, denoted by , to finance aggregate consumption by old adults (denoted by and by middle-aged adults (denoted by , and to cover costs of rearing children (denoted by , where represents average costs of rearing children per middle aged adult). Rearing children is a production activity that takes place within each household and its costs (per middle aged adult of every family) are determined by a non-decreasing convex function , satisfying

At any period, the aggregate resource constraint is

\[C _ {t} ^ {o} + C _ {t} ^ {m} + N _ {t - 1} b _ {t} ^ {m} + K _ {t + 1} \leq F _ {t} (K _ {t}, N _ {t - 1}),\tag{1}\]

which, by letting

\[c _ {t} ^ {o} = \left\{ \begin{array}{c l} C _ {t} ^ {o} / N _ {t - 2}, & \text {if} N _ {t - 2} > 0, \\ 0, & \text {otherwise}; \end{array} \right.\]

and

\[c _ {t} ^ {m} = \left\{ \begin{array}{c l} C _ {t} ^ {m} / N _ {t - 1}, & \text {if} N _ {t - 1} > 0, \\ 0, & \text {otherwise}; \end{array} \right.\]

can be equivalently written as

\[c _ {t} ^ {o} + n _ {t - 1} \left[ c _ {t} ^ {m} + b _ {t} ^ {m} + k _ {t + 1} ^ {o} \right] \leq F _ {t} (k _ {t} ^ {o}, n _ {t - 1}),\tag{2}\]

where represents average consumption per old adult, and represents average consumption per middle aged adult.

Throughout most of the paper, we will restrict attention on feasible symmetric allocations in which any two agents of the same generation who get to be alive take the same consumption and investment decisions; this allows one to identify per capita decisions with individual decisions. In particular, costs per middle aged adult of rearing children are given by , where is the number of children per middle aged adult. Thus, a feasible symmetric allocation will be represented by a sequence satisfying, for each the resource constraint

\[c _ {t} ^ {o} + n _ {t - 1} \left[ c _ {t} ^ {m} + b _ {t} (n _ {t}) + k _ {t + 1} ^ {o} \right] \leq F _ {t} (k _ {t} ^ {o}, n _ {t - 1});\tag{3}\]

and the initial condition

\[(n _ {- 2}, n _ {- 1}, k _ {0} ^ {o}) = \left(1, \overline {{n}} _ {- 1}, \overline {{K}} _ {0}\right).\tag{4}\]

Denote by the set containing all feasible symmetric allocations. For each agent born at , preferences on are represented by a utility function defined, for each , by where denotes the old adult’s consumption at period . For each agent born in period with preferences are represented by a utility function defined, for each , by , so that individuals may receive direct utility from consumption as well as the number of descendants they bear.⁹ We assume that the function is non-decreasing and continuously diferentiable, with indiference curves that are strictly convex with respect to the origin and do not cross the axis.

Ut₋₁(a) = u(c^m_t , c^o_t+1, nt) + βc^m_t+1

β

⁹Conde-Ruiz et al (2004) show that the main results are compatible with some kind of altruistic utility functions, in particular the family of utility functions , where β is a parameter representing the degree of ascending altruism.

Three final observations are in order. First, observe that if the population size of every generation is always positive, total output per worker (defined by can be written as a well defined function of capital per worker (defined by ; that is,

\[y _ {t} ^ {m} = F _ {t} (k _ {t} ^ {m}, 1) \equiv f _ {t} (k _ {t} ^ {m}),\]

and the resource constraint in (3) can be equivalently represented as

\[c _ {t} ^ {o} + n _ {t - 1} \left[ c _ {t} ^ {m} + b _ {t} (n _ {t}) + n _ {t} k _ {t + 1} ^ {m} \right] \leq n _ {t - 1} f _ {t} (k _ {t} ^ {m}).\]

Notice that this representation imposes, however, that for every allocation for which one necessarily has , agents cannot obtain resources for their old-age without having children. Thus, such representation seems inadequate to represent allocations for which for some t in environments where for some

Second, observe that each utility function represents the preferences of every agent efectively born at date t, but it does not provide any information on whether or not a potential agent prefers to be born at t and obtain a given bundle of goods rather than not to be born. In other words, the representation of the preference relation by the utility functions , with does not constitute a complete preference ordering of every potential agent on the set of feasible allocations.

Finally, note that the term in the left hand side of (3) is a quasiconcave function of the endogenous variables and . Due to this fact, the set of sequences satisfying the resource constraint in (3) and the initial condition in (4) is not a convex set, as it would if the sequence were fixed exogenously. As we show through the paper, this non-convexity will make it dificult to identify eficient allocations as defined in the following section.

3 Millian Efficient Allocations

The most commonly used optimality notion in standard normative economic analysis is that of Pareto eficiency. This notion of eficiency relies in turn on the well known Pareto criterium to compare social alternatives, a criterium that allows one to construct a partial ordering on the set of alternatives from the complete preference orderings (defined on this set) of a fixed group of agents. An eficient allocation can be described as a maximal element of the partial order induced by the Pareto criterium on the set of feasible allocations.

With endogenous populations, we can still use the Pareto criterium to rank feasible allocations using the partial orderings of all potential agents, represented by the utility functions of the living agents. That is, an allocation can still be ranked as Pareto superior to another one if it is unanimously preferred by all potential agents according to their partial preference ordering. However, this implies that any two allocations with diferent population size cannot be ranked, since we do not know whether or not an agent who lives in one allocation a but not in other allocation is better of in the latter than he is in the former. To avoid this problem and preserve the partial order induced by the Pareto criterium, one needs to extend it to compare also allocations of diferent population size.

A possible general extension of the Pareto criterium, applicable to any environment with endogenous fertility, can be constructed by ranking any two allocations making use of the Pareto criterium when the information of the preference profiles of those agents who are born in the two allocations is considered. This extension has recently been suggested also by Golosov, Tertilt and Jones (2004), who refer to it as the dominance criterium (where stands for alive agents). More precisely, the notion of dominance can be defined as follows.

Definition 1 For any two feasible allocations corresponding to an environment with endogenous fertility, a is said to dominate an allocation a^′ if a is unanimously preferred to a^′ by all agents who are born in both a and , and it is strictly preferred by some of these agents.

Observe that the dominance criterium is a general criterium, applicable to any two feasible allocations corresponding to an environment with endogenous population. The criterium can therefore be applied to rank any two allocations, although that would require a specification of the identity of every potential agent in the economy, which taking into account that there is a continuum of potential agents in every period, would involve considerable notational costs.

To avoid notational costs and make the dominance criterium suitable to undertake welfare comparisons of symmetric allocations without specifying the identity of every potential agent, we will adopt in what follows the following convention: for every two symmetric allocations a, for which the size of a given generation t is strictly positive (that is, such that and there exists a positive measure of agents born at t in the two allocations.

With this convention, the restriction of the relation induced by the dominance criterium to the set of symmetric allocations can be defined formally as follows.

Definition 2 A feasible allocation is said to dominate an allocation if

i) for every for which and one has

\[U _ {t} (a) \geq U _ {t} (a ^ {\prime});\]

and,

ii) there exists at least one period τ satisfying

\[\begin{array}{r c l} n _ {\tau} & > & 0, \\ n _ {\tau} ^ {\prime} & > & 0, a n d \\ U _ {\tau} (a) & > & U _ {\tau} (a ^ {\prime}). \end{array}\]

Thus, according to the dominance criterium, a symmetric allocation a dominates another one if it provides all agents living under the two allocations with the same welfare, and some of them with more utility.

The following example shows that even if we restrict its scope to compare only symmetric allocations, the notion of dominance brings with it an important dificulty: it induces a nontransitive relation on

Example 1. Non transitivity of the dominance relation. Consider a stationary economy described by a constant utility function , a constant production function and a constant cost function . Observe that the stationary allocation such that for gives all agents who get to be alive a utility level and for all

Consider now a date and an allocation such that

\[\left(\widetilde {c} _ {t} ^ {m}, \widetilde {c} _ {t + 1} ^ {o}, \widetilde {n} _ {t}, \widetilde {k} _ {t + 1} ^ {o}\right) = \left\{ \begin{array}{l l} (1, 1, 1, 1), & \text {if t = 0,1..., \tau - 1} \\ (2, 2, 0, 1), & \text {if t = \tau} \\ (0, 0, 0, 0), & \text {if t > \tau .} \end{array} \right.\]

Such allocation ields and

\[U _ {t - 1} (\widetilde {a}) = \left\{ \begin{array}{c l} 5, & \mathrm{if} t = 0, 1..., \tau - 1; \\ 4 \sqrt {2} > 5, & \mathrm{if} t = \tau ; \end{array} \right.\]

which taking into account that only those agents born at are alive in both and implies that a dominates But then let ebe an allocation such that

\[\left(\overline {{c}} _ {t} ^ {m}, \overline {{c}} _ {t + 1} ^ {o}, \overline {{n}} _ {t}, \overline {{k}} _ {t + 1} ^ {o}\right) = \left\{ \begin{array}{l l} (1, 1, 1, 1), & \text {if} t = 0, 1..., \tau - 1 \\ \left(\frac {3}{2}, 3, \frac {1}{2}, 1\right), & \text {if} t = \tau \\ (0, 0, 0, 0), & \text {if} t > \tau . \end{array} \right.\]

Such allocation yields and

\[U _ {t - 1} (\overline {{a}}) = \left\{ \begin{array}{c l} 5, & \text { if } t = 0, 1..., \tau - 1 \\ \sqrt {6} + 2 \sqrt {3} + \sqrt {\frac {1}{2}} > 4 \sqrt {2}, & \text { if } t = \tau \\ u (0) = 0, & \text { if } t = \tau + 1 \end{array} \right.\]

Hence, dominates However, it is not the case that a dominates since and e. Therefore the notion of dominance induces a non-transitive relation on .

It should be noticed that the type of inconsistency appearing in Example 1 is present only if the dominance criterium is used to compare allocations (like the allocation a in the example) efor which the economy collapses at a given date and no more individuals are born.10 If this type of allocations are ruled out as socially undesirable and one restricts the set of allocations that can be compared to the set formed by symmetric allocations such that for all such inconsistencies are no longer present. Observe that an allocation dominates an allocation if for all for all one has and this inequality is strict for some period t. Thus, the restriction of the dominance relation to the set is transitive and anti-symmetric, and therefore constitutes a partial ordering on

With this restriction, the dominance criterium gives rise to an eficiency criterium, which we refer to as Millian eficiency (or simply, , to identify the set of maximal elements of the partial order induced by the dominance criterium on the set . Formally, the notion of Millian eficiency can be defined as follows.

Definition 3 A feasible allocation is said to be Millian eficient if there does not exist another feasible allocation bsuch that:

i) for all . one has

\[U _ {t - 1} (a ^ {\prime}) \geq U _ {t - 1} (\widehat {a}); a n d\]

ii) there exists at least one period τ such that

\[U _ {\tau - 1} (a ^ {\prime}) > U _ {\tau - 1} (\widehat {a}).\]

Thus, if an allocation is Millian eficient, then there is no way to make all living agents of every generation better of without making some living agents of a generation worse of. Although some other authors have also used this criterium under the name of “Pareto optimality,”¹¹ we use a diferent name to make clear that it results from restricting a particular extension of the Pareto criterium to the set of feasible allocations.¹² Also, although other names (such as, for example, constrained eficiency) might be more informative of the normative principles underlying this notion of eficiency, we use the term Millian eficiency because it generalizes a notion of optimality, referred to as Millian optimality, which has been frequently used in the literature. Millian optimum is an allocation maximizing a function of the form for a strictly positive sequence of intergenerational weights Clearly, a Millian optimum must be a Millian eficient allocation, but the converse is not in general true. Since the set of feasible allocations is unbounded, Millian social welfare functions may be not well defined for many feasible paths, including some that are Millian eficient ones.

Ft(Kt, 0) = 0

Kt ∈ ℜ₊

¹⁰Since it has been assumed that utility functions of living agents never cross the axes, this type of inconsistencies cannot occur if the production function verifies for all .

¹¹See Raut (1992) and more recently, Michel and Wigniolle (2003).

In the following sections, we provide necessary and suficient conditions characterizing Millian eficient allocations.

3.1 Necessary conditions. Static eficiency.

For every allocation and every , write for the amount of physical resources at period t not devoted to feed the old generation, that is,

\[e _ {t} = c _ {t} ^ {m} + b _ {t} (n _ {t}) + k _ {t + 1} ^ {o}.\]

With this notation, the necessary conditions for Millian eficiency can be stated as follows.

Proposition 4 Every eficient allocation verifies for each

\[\begin{array}{r l r} {u \left(\widehat {x} _ {t}\right) = \max _ {(x _ {t}, k _ {t + 1} ^ {o}) \in \mathfrak {R} _ {+} ^ {4}} \Big \{u \left(x _ {t}\right)} & : & {c _ {t} ^ {m} + b _ {t} (n _ {t}) + k _ {t + 1} \leq \widehat {e} _ {t};} \\ & & {F _ {t + 1} (k _ {t + 1} ^ {o}, n _ {t}) - c _ {t + 1} ^ {o} \geq n _ {t} \widehat {e} _ {t + 1} \Big \}.} \end{array}\tag{5}\]

Proof. By contradiction. Suppose that is an eficient allocation, and suppose there exists a period τ for which the 4-upla bcorresponding to the allocation is not a solution to the b boptimization problem in (5). Select now a point bsatisfying the two constraints in (5) in such a way that e e is satisfied, and let a be the allocation obtained from by replacing the term b e by such point. Such allocation is feasible because b must verify and e e. Note that a has been constructed in e e esuch a way that it satisfies e efor all e band for , which implies that e b e e b bis not Millian eficient, a contradiction that establishes Proposition 4.

Notice that provides the maximum utility that an individual born at , endowed with bunits of physical resources, can obtain without diminishing the resources available for the bnext generation. In view of this we will adopt the terminology proposed by Balasko and Shell (1980), and we will refer to an allocations satisfying the necessary conditions in Proposition 4 as a statically allocation.

5,

¹²As we will discuss in section 5, this restriction has important implications.

¹³See Nerlove, Razin and Sadka (1987), Cigno (1992,2000) or Groezen, Leers and Medjam (2003). The name Millian social welfare function refers to the fact that they can also be justified from a form of utilitarianism, called average utilitarianism, often associated to John Stuart Mill (see Razin and Sadka, 1995, ch.5). This form of utilitarianism postulates that welfare judgments involving diferent generations should be independent of the population size of each generation.

Note that since we are restricting attention to allocations in , every Millian eficient allocation must verify for all t, which together with the assumption that the b b bindiference curves representing the agents’ preferences do not cross the axis, implies that every Millian eficient allocation a must be interior; that is, it must belong to the formed by all sequences satisfying , and for all Taking this into account, the following corollary makes use of the first order conditions associated to the sequence of optimization problems (5) in the statement of Proposition 4 to provide an equivalent representation of the necessary conditions for Millian eficiency.

Corollary 5 For every statically eficient allocation , there exists a sequence of strictly positive real numbers satisfying, for every

\[R _ {t + 1} = \frac {u _ {1} ^ {\prime} (\widehat {x} _ {t})}{u _ {2} ^ {\prime} (\widehat {x} _ {t})} = f _ {t + 1} ^ {\prime} \left(\widehat {k} _ {t + 1} ^ {m}\right) = \frac {\frac {\widehat {c} _ {t + 1} ^ {o}}{\widehat {n} _ {t}}}{b _ {t} ^ {\prime} (\widehat {n} _ {t}) - \frac {u _ {3} ^ {\prime} (\widehat {x} _ {t})}{u _ {1} ^ {\prime} (\widehat {x} _ {t})} + \widehat {k} _ {t + 1} ^ {m}}.\tag{6}\]

Observe that the term at the right hand side of (6) can be regarded as the rate of return to investments in children.¹⁴ Thus, Corollary 5 establishes that for any eficient allocation, the marginal rate of return to investments in children must be equal to the marginal rates of return to investments in physical capital (measured in per worker terms), that in turn must be equal to the marginal rate of substitution between current and future consumption.

Moreover, any vector satisfying the necessary conditions in (6) and the two conbstraints in the optimization problem in (5) is indeed a solution to that problem. To see this, let be implicitly defined by

\[F _ {t + 1} (\phi_ {t + 1} (y _ {t + 1} ^ {o}, n _ {t}), n _ {t}) = y _ {t + 1} ^ {o}.\]

That is, determines the amount of physical capital that combined with workers produces units of output. Notice that is a homogeneous of degree one, convex function. Also, by monotonicity of preferences, the two inequality constraints in the optimization problem (5) must be binding for any solution to the problem, and therefore these bconstraints can be represented equivalently as

\[c _ {t} ^ {m} + b _ {t} (n _ {t}) + \phi_ {t + 1} \left(c _ {t + 1} ^ {o} + n _ {t} \widehat {e} _ {t + 1}, n _ {t}\right) = \widehat {e} _ {t}.\tag{7}\]

Since is convex, it is straightforward to show that, given , the set of triples bsatisfying (7) is a convex set, which implies any vector satisfying the necessary bconditions in (6) and the feasibility condition in (7) corresponds to a solution to the optimization problem (5).

Remark 1. Recall that given a sequence of intergenerational weights, a Millian social welfare function is a function defined, for every , by . It is straightforward to show that any allocation maximizing a Millian bwelfare function among those feasible symmetric allocations must satisfy the necessary conditions in (6). However, even if is satisfied for an allocation verifying the first order conditions in (6), it is not in general true that such allocation maximizes on the set of feasible allocations . As we mentioned above and will become clear along the paper, the set is non-convex, and therefore first order conditions might be not suficient for a maximum. For this reason, the previously mentioned result stating that a Millian optimum must be a Millian eficient allocation is not particularly useful, since it says nothing on whether or not an allocation verifying the first order conditions of a Millian optimization problem is indeed a Millian optimum.

Rt+1 = u1(xbt)u′ (xt) = f′t+1 kot+1 bnt = cbt+1nb t′ + kot+1 b′t(nt) − bu′1(xt) n t

b b14Of course, necessary conditions in (6) can be equivalently written, in terms of capital per family, as

3.2 Suficient conditions. Dynamic eficiency.

In the previous subsection we have obtained necessary conditions for Millian eficiency, but those conditions do not guarantee that an allocation solving the sequence of optimization problems in the statement of Proposition 4 is actually eficient. As in other overlapping generation models, an allocation might solve the sequence of optimization problems in (5) and fails to be eficient. In the literature, this issue has been referred to as the dynamic eficiency problem.

Well known theoretical work deals with the issue of dynamic eficiency, such as Cass (1972) in the context of a simple physical capital growth model, and Balasko and Shell (1980) who focus on an overlapping generations exchange economy. All these papers show that, despite an allocation can be short-run eficient (or statically eficient), i.e., it cannot be improved upon by a reallocation of resources of a finite number of generations, it might not be long-run eficient (or dynamically eficient), that is, fully eficient. In this subsection, we extend these previous results to an environment of endogenous population.

Initially, we introduce some notation. Let and b b b b b b b bbe a solution to the system of equations given by first order conditions in (6) b band the feasibility condition in (7). Note that is a solution to the optimization problem (5) in the statement of Proposition 4.

Now, let , the indirect utility function of an agent belonging to generation , be defined, for every and every pair , by

\[W _ {t - 1} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) = u \left(x _ {t} (\widehat {e} _ {t}, \widehat {e} _ {t + 1})\right).\tag{8}\]

That is, for each , the function at each determines the maximum utility b bachievable by an agent born at time t 1 with the (per worker) resources of the homogeneous good given by and at least units of the homogeneous good provided from the resources b bproduced by the following generation.15

Notice also that each indirect utility function is strictly increasing in , strictly decreasing in , and continuously diferentiable on the interior of its domain. Therefore the slope of the indiference curve passing through an arbitrary point is well defined b bas long as b bis well defined. The Implicit Function Theorem yields

\[m (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) = - \frac {\frac {\partial W _ {t - 1} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}{\partial e _ {t}}}{\frac {\partial W _ {t - 1} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}{\partial e _ {t + 1}}} = - \frac {\lambda_ {t} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}{\lambda_ {t + 1} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})},\]

where and are the Kuhn-Tucker multipliers for which the first order b b b bconditions of the optimization problem (5) are satisfied. Taking this into account we obtain, by the Envelope Theorem,

\[\lambda_ {t} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) = u _ {1} ^ {\prime} (x _ {t} (\widehat {e} _ {t}, \widehat {e} _ {t + 1})),\]

a ∈ S^I

Ut₋₁(a) = u (xt) = Wt₋₁(et, et₊₁)

¹⁵Note that for every static M−eficient allocation one must have .

and

\[\lambda_ {t + 1} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) = - u _ {2} ^ {\prime} (x _ {t} (\widehat {e} _ {t}, \widehat {e} _ {t + 1})) n _ {t} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}).\]

Therefore,

\[m _ {t} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) = \frac {R _ {t + 1} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}{n _ {t} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}.\]

bRecall that for a statically eficient allocation bwith each term b bprovides the maximum utility that an agent born at any period can obtain without afecting the utility derived by the next generation. Consequently, the issue of whether is dynamically eficient can be reduced to a question on whether or not a sequence bthat eimproves the indirect utility of an infinite number of generations exists. Next we present a result, similar to Balasko and Shell (1980, Lemma 5.4), that will be useful to dismiss allocations that are statically eficient but not dynamically eficient. It establishes that improving upon a statically eficient allocation imposes that total resources available for old agents must be increased at every period.

Lemma 6 Let be an allocation satisfying the necessary conditions , and suppose is bineficient. Then there exists an allocation bthat Millian dominates the allocation and some period , such that,

\[\widetilde {e} _ {t} \leq \widehat {e} _ {t} f o r a l l t \geq 0, a n d \widetilde {e} _ {\tau} < \widehat {e} _ {\tau} f o r a l l \tau \geq T.\tag{9}\]

Proof. Let be an ineficient allocation satisfying the conditions , and let be an ballocation that Millian dominates the allocation , that is, an allocation satisfying

\[W _ {t - 1} (\widetilde {e} _ {t}, \widetilde {e} _ {t + 1}) \geq W _ {t - 1} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) \text { and } W _ {T - 1} (\widetilde {e} _ {T}, \widetilde {e} _ {T + 1}) > W _ {T - 1} (\widehat {e} _ {T}, \widehat {e} _ {T + 1}) \text { for some } t = T.\]

To show satisfies condition (9), observe first that verifies , where the inequality emust be strict if e. Taking into account that b is strictly increasing in one obtains

\[W _ {- 1} (\widetilde {e} _ {0}, \widehat {e} _ {1}) \leq W _ {- 1} (\widehat {e} _ {0}, \widehat {e} _ {1}).\]

Also, since is strictly decreasing in , the inequality is only satisfied if e, where the last inequality must be strict if either or e bis satisfied. Proceeding analogously, since e eis strictly decreasing in b band the e binequality must be satisfied one must have (with if either or e b e bholds). By applying the argument recursively one obtains

\[\widehat {e} _ {t} - \widetilde {e} _ {t} \geq 0 \mathrm{forall} t \geq 0\]

and

\[\widehat {e} _ {\tau} - \widetilde {e} _ {\tau} > 0 \text { for some } T \text { and all } \tau \geq T,\]

which establishes condition (9) and, therefore, completes the proof of Lemma 6.

With the properties of the function in mind, the problem of dynamic eficiency is analyzed below.

3.2.1 The stationary case. We will first gain some intuitions by considering stationary allocations (that is, allocations such that for all in an economy with no technological progress (i.e., such that and for all . Note that in this case one has and for all , that is, the indirect utility function and the function determining the slope of the indiference curve passing through any point are the same for all generations. Since for any such stationary allocations one has for all , the set of stationary allocations is represented by the line in

To simplify things, assume W is strictly quasiconcave, that is, the slope of any indiference curve (which is given by decreases as increases. Consider now a point like in Figure 1, corresponding to an allocation satisfying the necessary conditions in (6), with . Note that for such allocation one has . Clearly, such allocation is not eficient since by reducing towards the point in the figure, all agents are better of. By contrast, consider now a point like corresponding to an allocation for which . Apparently, it is possible to improve all agents by increasing in the direction of . However, achieving Pareto improvements by increasing is impossible, because increasing in period implies that agents born at time ., the old generation at period necessarily decrease their consumption and, hence, their utility. Thus, such an allocation cannot be dominated by any other stationary allocation.

Thus, if the indirect utility function is strictly quasiconcave, then all allocations a^′ for which , all ineficient allocations) verify whereas all allocations for which (that is, all stationary allocations that are not dominated by any other stationary allocation) verify the Phelps-Koopmans-Diamond K-D] condition (see Galor and Ryder, . The stationary allocation for which verifying has been referred to as the golden rule allocation, since it maximizes the utility obtained by a representative agent among those feasible stationary allocations.

If the indirect utility function is not quasiconcave, condition is still valid to conclude that a statically eficient stationary allocation a is not dynamically eficient. However, condition no longer guarantees that a statically eficient stationary allocation a is also dynamically eficient. This case is illustrated in Figure 2, in which a point corresponding to an allocation verifying such condition is not an eficient allocation. e eObserve that the line with slope passing through does not separate the upper contour e e e eset of this point. Note that the steepest hyperplane passing through the point separating the upper contour set of e eis given by the dotted line in the figure. In what follows we use the e eslope of this line to provide an alternative criterium ensuring that an allocation is eficient.

Given a pair , define

\[\pi_ {t} \left(\widehat {e} _ {t}, \widehat {e} _ {t + 1}\right) = \inf \left\{\frac {\widehat {e} _ {t + 1} - e _ {t + 1}}{\widehat {e} _ {t} - e _ {t}}: (e _ {t}, e _ {t + 1}) < < (\widehat {e} _ {t}, \widehat {e} _ {t + 1}), W _ {t - 1} (e _ {t}, e _ {t + 1}) \geq W _ {t - 1} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) \right\}.\]

That is, is the steepest slope of the hyperplane passing through a point b b bseparating the set of allocations that improve welfare of agents belonging to generation Notice that,

\[\pi_ {t} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) \leq \frac {\widehat {R} _ {t + 1} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}{\widehat {n} _ {t} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})} = m _ {t} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}),\]

¹⁶These authors attribute the condition to Phelps (1965) and Koopmans (1965) in the context of growth models with exogenous saving rate. Diamond (1965) extended this result to an overlapping generations model with production.

where the inequality above holds as a strict equality whenever is quasiconcave.

With this notation, the set of stationary allocations that are not Millian dominated by any other stationary allocation is characterized as follows.

Lemma 7 Let be a stationary allocation for which for all t. Then is not bdominated by any other stationary allocation if and only if

\[\pi_ {t} (\widehat {e}, \widehat {e}) = \pi (\widehat {e}, \widehat {e}) \geq 1.\tag{10}\]

The proof is straightforward from the definition of and Lemma 6.

3.2.2 The general case. Cass (1972) and Balasko and Shell (1980) have ofered suficient conditions for dynamic eficiency that can be applied to non-stationary allocations. However, these conditions may not be suficient any longer due to the non-convexity problem described above. The following proposition provides a suficient condition that guarantees that an allocation is eficient in a general economic environment, even if the indirect utility function fails to be quasiconcave.

Proposition 8 Consider an allocation satisfying the necessary condition (6). If

\[\lim _ {T \to \infty} \left(\frac {\widehat {e} _ {T}}{\prod_ {t = 0} ^ {T} \pi_ {t} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}\right) = 0,\tag{11}\]

then is Millian eficient.

Proof. Consider an allocation satisfying conditions (6), and (11),and suppose now that bit is not eficient. To show that this yields a contradiction, let a be an allocation dominating the allocation a, and let τ be the first period for which . Observe ethat by Lemma e emust be satisfied, and therefore there exists . Since a e bsatisfies condition (11), there must exist a suficiently large bsuch that, for each bone has

\[\left(\frac {\widehat {e} _ {T}}{\prod_ {t = 0} ^ {T} \pi_ {t} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}\right) < \epsilon = \widehat {e} _ {\tau} - \widetilde {e} _ {\tau}.\]

Use now condition (9) in the statement of Lemma 6 and the definition of to obtain the chain of inequalities

\[\begin{array}{r c l} 0 & < & (\widehat {e} _ {\tau} - \widetilde {e} _ {\tau}) = \epsilon \leq \frac {(\widehat {e} _ {\tau + 1} - \widetilde {e} _ {\tau + 1})}{\pi_ {\tau} (\widehat {e} _ {\tau} , \widehat {e} _ {\tau + 1})} \leq \frac {\widehat {e} _ {\tau + 2} - \widetilde {e} _ {\tau + 2}}{\pi_ {\tau} (\widehat {e} _ {\tau} , \widehat {e} _ {\tau + 1}) \pi_ {\tau + 1} (\widehat {e} _ {\tau + 1} , \widehat {e} _ {\tau + 2})} \leq \\ & \leq & \dots \leq \\ & \leq & \frac {\widehat {e} _ {T} - \widetilde {e} _ {T}}{\prod_ {t = \tau} ^ {T} \pi_ {t} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})} < \frac {\widehat {e} _ {T}}{\prod_ {t = \tau} ^ {T} \pi_ {t} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}, \end{array}\]

which contradicts condition (11) and, therefore, establishes that is Millian eficient.

Observe that despite no assumption on fundamentals guarantees the quasiconcavity of the indirect utility function W, though this property would allow one to simplify condition (11) as

\[\lim _ {T \to \infty} \left(\widehat {e} _ {T} \prod_ {t = 0} ^ {T} \frac {n _ {t} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}{R _ {t + 1} (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}\right) = 0,\]

which in the case that the allocation is stationary reduces to the P-D-K condition.

Apparently, the suficient condition in (11) is not very useful, since one needs to compute all terms in the sequence . However, the following corollaries simplify the criterium b bfor special cases. The first corollary indicates that if the sequence converges to a steady state then condition (11) reduces to

Corollary 9 Let be a statically eficient allocation converging to a stationary allocation efor which . Then, is Millian eficient.

Proof. Observe first that given that is a continuous function, for all t, then if converges to a stationary point , then the sequence of slopes will also converge to a stationary slope . This means that condition (11)

\[\lim _ {T \to \infty} \left(\widetilde {e} _ {T} \frac {1}{\prod_ {t = 0} ^ {T} \pi_ {t} (\widetilde {e} _ {t} , \widetilde {e} _ {t + 1})}\right) = 0\]

holds, as long as it is the limit of the product of two convergent sequences: that converges to , and the sequence that converges to

As we will show in Example 3, even in the stationary case computing the slope might be a dificult task. The following corollary provides an alternative criterium to identify eficient stationary allocations and, in view of the previous result, eficient allocations, non necessarily stationary.

Corollary 10 Let be a stationary allocation for which for all t, and suppose that for all for which b bis well defined. Then, the stationary allocation verifies and, therefore, it is Millian eficient.

Proof. Observe that, given that is increasing in and decreasing in , any stationary point below belongs to an indiference curve that provides strictly lower welfare; that is, b bgiven that (e, e) belongs to an indiference curve , then any with verifies 2 with b b b e e e b. Consequently, the slope of the hyperplane defined above passing through ewill e bbe strictly larger than b b. Then Proposition 8 implies that a is Millian eficient..

3.2.3 Examples. To conclude the section, we present two examples that illustrate how to apply the suficient condition provided in Proposition 8 to determine whether a given allocation is eficient or not. The first one refers to afine technologies, where the marginal productivity of capital is constant and therefore independent of its level. This simplified set-up helps to clarify the intuition of our suficient condition, and shows that the P-D-K criterium might erroneously regard dynamically ineficient allocations as being eficient. To the extent that such linear technologies can be seen as reduced form representations of technologies arising in small open economies (or exchange economies), such as those studied by Cigno (2003) or Groezen et al (2003), the example suggests that conclusions at which these authors arrive might be incorrect.

Example 2. Linear Technology. Consider the following parametrization for preferences, technology and cost function for rearing children: a Cobb-Douglas utility function a linear production function, , with and a linear cost function,

For any economy in this class and the feasibility constraint in (5), equation (7), reduces to . Given that preferences are strictly monotone on b, the domain of definition in the problem (5) at each is restricted to sat isfy for every eficient allocation . The unique solution to the optimization bproblem in the definition of is bConsequently, the indirect utility function is in fact a b b b bstrictly quasiconvex function, whose indiference curves are convex to the origin, and given by

\[\mathcal {I} _ {w} = \left\{\left(\widehat {e} _ {t}, \widehat {e} _ {t + 1}\right) \in \Re_ {+} ^ {2}: \widehat {e} _ {t + 1} + b R - \omega > 0, \widehat {e} _ {t + 1} = \frac {R ^ {2} \widehat {e} _ {t} ^ {3}}{2 7 \omega} + w - b R, w > 0 \right\}.\]

In order to analyze dynamic eficiency of a given allocation, it is useful to distinguish between two possible cases, depending on whether the term is positive or not. First, if is satisfied, then all indiference curves passing through any stationary along the line, will cross once the at b b b. In consequence, given that any feasible point cannot be negative , and the indiference curves are convex, the steepest hyperplane defined in the previous section equals (see Figure 3). In view of Proposition b b b bany statically eficient allocation for which the sequence converges to must be Millian eficient. Observe that the P-D-K criterium also holds, i.e., b for any stationary allocations b, and consequently Corollary 10 is verified.

b bSecond, in the case that the domain of the indirect utility function is restricted by . Now, all indiference curves passing through any stationary would cross once the at b b, a point that is out of the domain, and does not cross the . Since the indiference curves are convex, and that for any feasible point, the steepest slope equals at any stationary (see Figure 4). This means b b b b b bthat any stationary allocation a is ineficient. Interestingly, the P-D-K criterium fails to provide this result at all.

In the second example, we show that under conditions of Corollary 10 it is plausible that all stationary allocations satisfying the necessary conditions are indeed . We explore the (dynamic) eficiency properties of an allocation by obtaining first an explicit expression for the sequence of indirect utility functions . Next, we show that despite the indirect utility function is not quasiconcave all stationary allocations will be statically eficient.

Example 3. Cobb-Douglas Technology. Consider the following parametrization: a Cobb-Douglas utility function a constant returns to scale technology, ; and a linear cost function, . For this economy, the feasibility constraint (7) is

\[c _ {t} ^ {m} + n _ {t} \left(b + \left(\widehat {e} _ {t + 1} - (c _ {t + 1} ^ {o} / n _ {t})\right) ^ {2} / 4\right) = \widehat {e} _ {t}.\]

First order conditions provides the unique solution to the optimization problem

\[\left(\widehat {c} _ {t} ^ {m}, \widehat {c} _ {t + 1} ^ {o}, \widehat {n} _ {t}\right) = \left(\widehat {e} _ {t} / 3, (2 / 3) \widehat {e} _ {t} / [ \widehat {e} _ {t + 1} + \Lambda (\widehat {e} _ {t + 1}) ], 2 \widehat {e} _ {t} / [ 3 (\widehat {e} _ {t + 1} + \Lambda (\widehat {e} _ {t + 1})) \Lambda (\widehat {e} _ {t + 1}) ]\right),\]

and , with . The indirect b butility function bis

\[W _ {t - 1} (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) = \frac {\widehat {e} _ {t} ^ {3} \Big ((3 b + \widehat {e} _ {t + 1} ^ {2}) ^ {1 / 2} - \widehat {e} _ {t + 1} \Big) ^ {2} \Big (2 (3 b + \widehat {e} _ {t + 1} ^ {2}) ^ {1 / 2} + \widehat {e} _ {t + 1} \Big)}{2 7 b ^ {2} (4 b + \widehat {e} _ {t + 1} ^ {2})}.\]

Given that the slope of the indiference curves are

\[m (\widehat {e} _ {t}, \widehat {e} _ {t + 1}) = \frac {R (\widehat {e} _ {t} , \widehat {e} _ {t + 1})}{n (\widehat {e} _ {t} , \widehat {e} _ {t + 1})} = \frac {\widehat {e} _ {t + 1}}{\widehat {e} _ {t}} \left[ 2 \Big (1 + \frac {3 b}{\widehat {e} _ {t + 1}} \Big) ^ {1 / 2} - 1 \right] > 1,\]

it is straightforward to show that for all . This implies, by Corollary 10, that for all b b b b, that is, all the stationary allocations are Millian eficient.

4 A characterization of efficient allocations as decentralized equilibria

In this section, we show that every allocation satisfying the necessary conditions in Corollary 5 can be characterized as the outcome of a decentralized sequential price mechanism described as follows:

At each date , three markets (to which all agents have free access) are open: a spot good market where the resources produced in the economy are demanded by adult agents; a financial market, that allows agents to lend (or borrow) one unit of the homogeneous good in period t and obtain (or pay back) units of the same good in period ; and, a spot job market, in which labor is exchanged against the homogeneous good at a price

In addition to these external markets, there exists a sequence of intergenerational contracts, represented by a sequence of fees that obliges all agents born at date t to pay to their parents a fee equal to units of the consumption good when they reach their middle-aged reported at . The payment of this fee can be thought of as an agents’ compensation to their parents for the costs in which parents incurred in order to make it possible that these agents were born.

With these three markets operating at each date, the life-cycle optimization problem for an agent born in period , with is

\[\begin{array}{r c l} V _ {t - 1} (w _ {t} - \rho_ {t}, R _ {t + 1}, \rho_ {t + 1}) & = & \max _ {(x _ {t}, s _ {t})} u (c _ {t} ^ {m}, c _ {t + 1} ^ {o}, n _ {t}) \\ & & \text {subject to:} c _ {t} ^ {m} + s _ {t} + b _ {t} (n _ {t}) \leq w _ {t} - \rho_ {t} \\ & & c _ {t + 1} ^ {o} \leq R _ {t + 1} s _ {t} + n _ {t} \rho_ {t + 1}; \end{array}\tag{12}\]

given and ; and where represents the middle-aged agent’s savings at period t.

The profit maximization problem faced by the aggregate firm, where inputs are purchased at competitive prices, is

\[\begin{array}{r c l} \Pi_ {t} (w _ {t}, R _ {t}) & = & \max _ {(K _ {t}, N _ {t - 1})} \left\{F _ {t} (K _ {t}, N _ {t - 1}) - w _ {t} N _ {t - 1} - R _ {t} K _ {t} \right\} \\ & = & \max _ {(k _ {t} ^ {m}, N _ {t - 1})} N _ {t - 1} \left\{f _ {t} (k _ {t} ^ {m}) - w _ {t} - R _ {t} k _ {t} ^ {m} \right\}. \end{array}\]

for each period , and given the initial stock of capital . The notion of a decentralized equilibrium generated by a sequence of fees can be defined formally as follows:

Definition 11 For a given sequence of contracts a decentralized equilibrium (generated is an allocation and a sequence of prices such that for each

i) agents born at choose their consumption , their savings , and the number of descendants to solve the maximization problem in (12);

ii) the aggregate firm choose labor and capital per worker to maximize their profits, that is, and ; and,

iii) markets clear at all dates, that

\[s _ {t} ^ {*} = n _ {t} ^ {*} k _ {t + 1} ^ {m *},\]

or, equivalently,

\[c _ {t} ^ {o *} + n _ {t - 1} ^ {*} \left[ c _ {t} ^ {m *} + b _ {t} (n _ {t} ^ {*}) + s _ {t + 1} ^ {*} \right] = n _ {t - 1} ^ {*} f _ {t} (k _ {t} ^ {m *}).\]

Thus, an interior decentralized equilibrium associated to a sequence of fees can be equivalently characterized as an interior allocation for which there exists a sequence of prices such that, for each t, the following conditions are satisfied:

\[\begin{array}{r c l} \frac {u _ {1} ^ {\prime} (x _ {t} ^ {*})}{u _ {2} ^ {\prime} (x _ {t} ^ {*})} & = & R _ {t + 1} ^ {*}; \\ \left[ b ^ {\prime} (n _ {t} ^ {*}) - \frac {u _ {3} ^ {\prime} (x _ {t} ^ {*})}{u _ {1} ^ {\prime} (x _ {t} ^ {*})} \right] \frac {u _ {1} ^ {\prime} (x _ {t} ^ {*})}{u _ {2} ^ {\prime} (x _ {t} ^ {*})} & = & \rho_ {t + 1}; \\ R _ {t + 1} ^ {*} & = & f _ {t + 1} ^ {\prime} (k _ {t + 1} ^ {m *}); \\ w _ {t + 1} ^ {*} & = & f _ {t + 1} (k _ {t + 1} ^ {m *}) - f _ {t + 1} ^ {\prime} (k _ {t + 1} ^ {m *}) k _ {t + 1} ^ {m *}; \end{array}\]

and finally, the market clearing condition

\[c _ {t} ^ {o *} + n _ {t - 1} ^ {*} \left[ c _ {t} ^ {m *} + b (n _ {t} ^ {*}) + s _ {t + 1} ^ {*} \right] = n _ {t - 1} ^ {*} f _ {t} (k _ {t} ^ {m *});\tag{13}\]

Observe that the second condition holds for any fees pattern, including . These equations provide a straightforward characterization of statically eficient interior allocations as the equilibria of the sequential price mechanism described above, as the following result states.

Theorem 12

(i) Consider an arbitrary sequence of intergenerational contracts, and let be an interior decentralized equilibrium generated by . Then is statically

(ii) For every statically cient interior allocation , there exists a sequence of intergenerational contracts generating as a decentralized equilibrium.

Proof. To prove , let be an interior decentralized equilibrium generated by a sequence and let be the sequence of wages and interest rates corresponding to such equilibrium. Using (13), it is straightforward to check that is a feasible allocation satisfying, for each

\[R _ {t + 1} ^ {*} = \frac {u _ {1} ^ {\prime} (x _ {t} ^ {*})}{u _ {2} ^ {\prime} (x _ {t} ^ {*})} = f _ {t} ^ {\prime} (k _ {t + 1} ^ {m *}).\]

Thus, in order to show that satisfies the necessary conditions in (6) (and, hence, that it is statically eficient), we simply need to show that

\[R _ {t + 1} ^ {*} = \frac {c _ {t + 1} ^ {o *} / n _ {t} ^ {*}}{b ^ {\prime} (n _ {t} ^ {*}) - \frac {u _ {3} ^ {\prime} (x _ {t} ^ {*})}{u _ {1} ^ {\prime} (x _ {t} ^ {*})} + k _ {t + 1} ^ {m *}}\tag{14}\]

is satisfied. To prove it, make use the constraint for the agent born at t 1 when old adult in (12), the condition , and the market clearing condition in capital markets to write the term at the right hand side of (14) as

\[\frac {c _ {t + 1} ^ {o *} / n _ {t} ^ {*}}{b ^ {\prime} (n _ {t} ^ {*}) - \frac {u _ {3} ^ {\prime} (x _ {t} ^ {*})}{u _ {1} ^ {\prime} (x _ {t} ^ {*})} + k _ {t + 1} ^ {m *}} = \frac {\rho_ {t + 1} + \frac {s _ {t} ^ {*}}{n _ {t} ^ {*}} R _ {t + 1} ^ {*}}{\frac {\rho_ {t + 1}}{R _ {t + 1} ^ {*}} + \frac {s _ {t} ^ {*}}{n _ {t} ^ {*}}} = R _ {t + 1} ^ {*},\]

which establishes (i).

To prove (ii), let be an interior, statically eficient allocation and be the rate of return implicitly defined by the necessary conditions in (6). Then using the equilibrium conditions in (13), it is straightforward to check that a sequence of intergenerational transfers defined by generates as an interior decentralized equilibrium, which completes the proof of the statement (ii) in Theorem 12.

Thus, the notion of Millian eficiency admits a characterization that is closely analogous to the one provided by the two Fundamental Theorems of Welfare Economics. Nevertheless, two important diferences arise.

With respect to the statement (i) in Theorem 12, that can be regarded as a version of the First Fundamental Welfare Theorem, it is important to observe that the equilibrium generated by a sequence of contracts for which for all t is statically eficient. Although other authors have reached to similar conclusions, such as Nerlove, Razin and Sadka (1985) in a twoperiod framework or Groezen, Leers and Medjam (2003) in a infinite-period setting, they both identify an eficient allocation with a maximum of a Millian social welfare function. In the case of Groezen et al (2003), their version of the statement (i) in Theorem 12 applies only to a parametric example of a stationary economy, and it is required that a given parameter determining the agents preferences equals the (constant) intergenerational discount factor in the definition of the Millian social welfare function. Therefore their result is not robust to changes in the specifications of the agents’ preferences.

Both Nerlove, Razin and Sadka (1985) and Groezen et al (2003) interpret such equilibria as the natural, laissez faire equilibrium that would arise in an economy with free access to capital markets, since any other contract between parents and their ofspring would be time inconsistent and children cannot sign contracts on the right to be born. In contrast with this view, Erhlich and Lui (1991) and Cigno (1993, 2003) have argued that intergenerational transfers from children to their parents might play a role in sustaining, as subgame perfect equilibria of implicit contracting games, intrafamiliar contracts that allow children to finance their human capital accumulation. In view of this, we might alternatively consider the equilibria generated by as an equilibrium with missing markets, since there is no market in which parents and their ofspring can bargain on the right to exist. In this sense, this version of the First Theorem of Welfare Economics contrasts with well known results on welfare properties of economies with missing markets. Here, the absence of a market yields no eficiency loss, at least if one regards eficiency as Millian eficiency.

The statement (ii) of Theorem 12 can be regarded as a version of the Second Fundamental Theorem of Welfare Economics. Similar to the case of economies with exogenous population, every Millian eficient allocation can be decentralized by initially selecting an appropriate sequence of intergenerational transfers, and then allowing the agents to determine their consumption and investment decisions at competitive markets. Diferently from the standard, exogenous population case, in which non-distorting intergenerational transfers must be lump-sum for all agents, an incentive scheme that links intergenerational transfers with fertility decisions is needed. More precisely, for every system of intergenerational transfers that achieves Millian eficiency, every middle-aged adult has to pay a tax (or, in some cases, receive a lump-sum subsidy given by , while every old adult has to receive a subsidy (or pay a tax) which depends linearly of the number of children she decides to have.

In a nutshell, the first part of Theorem 12 shows that intergenerational transfers determined through competitive financial markets might be suficient to achieve an eficient allocation. However, the second part shows that if we intend to correct the intergenerational transfers determined through competitive financial markets, then we have to design a set of transfer policies which are not lump-sum but depend on the individuals’ decisions (the number of descendants).

We should also point out that the characterization given above refers to statically eficient allocations. Of course, many of the equilibria described above might be dynamically ineficient, and therefore it is worth exploring what type of intergenerational contracts ensures that dynamic eficiency is achieved. A useful way to answer this question is by representing the equilibrium path corresponding to a sequence of intergenerational transfers as a sequence satisfying a diference equation of the form

\[e _ {t + 1} ^ {*} = H _ {t} (e _ {t} ^ {*}, \rho_ {t + 1}).\]

In the context of the economies studied in Examples 2 and 3, it is possible to obtain an explicit expression of the function H characterizing such equilibrium path. Furthermore, in these two cases, each function is independent of which implies that the equilibrium path corresponding to a sequence of contracts adopts the form

\[e _ {t + 1} ^ {*} = H (\rho_ {t + 1});\]

that is, in the context of the specific economies described in our Examples 2 and 3, the equilibrium path corresponding to a stationary sequence of contracts such that for all t achieves a steady state in one period.

5 Alternative notions of efficiency with endogenous fertility

In this section, we provide alternative extensions of the notion of Pareto eficiency to the environment studied throughout the paper, and explore the robustness of the Millian notion of eficiency by exploring whether or not some Millian eficient allocations are also eficient under these alternative notions.

5.1 u dominance and constrained u eficiency

The first notion of eficiency explored in this section results from applying an alternative extension of the Pareto criterium, referred to as the criterium, to the set of feasible allocations. The dominance criterium is obtained from a previous assumption on the utility level obtained by non-born agents, which together with the utility functions of all living agents provides a complete description of preferences of all potential agents in the economy across all social states. To be more precise, we assume that a (potential) agent would weakly prefer being born at t if she obtains utility rather than not being born (and therefore obtain u); and that an agent would weakly prefer not to be born at t rather than being born at and obtain

This assumption provides a straightforward extension of the Pareto dominance criterium to compare allocations of diferent population size.

Definition 13 For any two feasible allocations a, a^′ corresponding to an environment with endogenous fertility, a is said to u-dominate an allocation a^′ if a is unanimously preferred to a^′ by all potential agents in the economy, and strictly preferred by some of these agents, provided the utility obtained by non-born agents is given by u.

Thus, the u dominance criterium is formed by directly applying the Pareto criterium to rank social alternatives using information on the complete preference orderings of all potential agents in the economy, represented by the sequence of the utility functions of the living agents and the utility threshold We regard the threshold utility level u not as the true utility level of a nonborn agent, but as one way of capturing social judgements determining under what circumstances it is worth living.¹⁸ For example, a social judgement according to which living is always worth can be represented by a threshold such that

As with the notion of dominance, the restriction of the relation induced by the u dominance criterium to the set of allocations gives rise to a notion of eficiency, which we will refer to as constrained , that can be defined as follows.

Definition 14 A feasible allocation is said to be ^∗ constrained if there does not exist another feasible allocation such that:

i) for all . one has

\[\begin{array}{r c l} {U _ {t} (a ^ {\prime})} & \geq & {U _ {t} (\widehat {a});} \\ {n _ {t} ^ {\prime}} & \geq & {n _ {t} w h e n e v e r U _ {t} (a ^ {\prime}) \geq U _ {t} (a) \geq \underline {{u}}, a n d} \\ {n _ {t} ^ {\prime}} & \leq & {n _ {t} w h e n e v e r \underline {{u}} > U _ {t} (a ^ {\prime}) \geq U _ {t} (a);} \end{array}\]

and,

ii) there exists at least one period τ for which at least one of the above conditions holds with inequality, that is.

\[\begin{array}{r c l} {U _ {\tau} (a ^ {\prime})} & > & {U _ {\tau} (\widehat {a}); o r} \\ {n _ {\tau} ^ {\prime}} & > & {n _ {\tau} a n d U _ {\tau} (a ^ {\prime}) \geq U _ {\tau} (a) \geq \underline {{u}}, o r} \\ {n _ {\tau} ^ {\prime}} & < & {n _ {\tau} a n d \underline {{u}} > U _ {\tau} (a ^ {\prime}) \geq U _ {\tau} (a).} \end{array}\]

It should be noticed that an constrained u eficient allocation may not be eficient in the Millian sense, since increasing utility of some generation might require to alter the number of individuals of some other generation. Nevertheless, observe that conditions establishing u dominance among interior allocations are stronger than those guaranteeing dominance, and therefore every Millian eficient allocation is also ^∗ constrained u eficient.

We should also point out that the notion of u dominance does not give rise to the consistency problems afecting the dominance relation. Thus, ruling out all symmetric allocations for which for some t on is unnecessary from that point of view. Without that restriction, an allocation a satisfying for some and every is u dominated by an allocation such that for all . Nevertheless, a Millian eficient allocation for which is still undominated by any other allocation in

¹⁷Recently, Golosov, Jones and Tertilt (2004) have provided a similar notion, which they call P−eficiency. In their view, however, the utility obtained by non-born agents, which might depend on the utility obtained by living agents, should be part of the specification of the model.

¹⁸Of course, any attempt for determining this threshold u constitutes an extremely dificult task, since we cannot rely on the market to obtain that type of information. We would like to point out, however, that being aware of these dificulties does not mean that the individuals forming a society should not decide under what circumstances it is worth living. Nevertheless, any justification of a particular value for u goes beyond the scope of this paper.

To summarize, the notion of Millian eficiency can be justified also on the grounds on an alternative criterium, the u dominance criterium, to compare allocations with diferent population size.

5.2 Allowing for Asymmetric Allocations: eficiency and

As we made clear, the notion of Millian eficiency (and constrained results from restricting an extension of the Pareto criterium to a certain class of symmetric allocations. With exogenous population, restricting the analysis to symmetric allocations is reasonable not only for reasons of analytical tractability or because symmetry is an appealing normative property, but also because any symmetric allocation that is not Pareto dominated by any other symmetric allocation is also eficient under a more general, unconstrained sense. That is, imposing symmetry does not involve eficiency losses.

In this section, we explore whether or not this is also true for the two extensions of the Pareto criterium analyzed in previous sections; that is, we analyze whether or not a symmetric, Millian eficient allocation may be dominated (or u dominated) by a non-symmetric allocation and therefore can be regarded as eficient (or 19

In order to explore this possibility and to simplify things, we will restrict attention to stationary Millian eficient allocations in and on a particular class of asymmetric allocations, which we denote by . For each allocation in this class, there are two types (or dynasties) of individuals, indexed by . Individuals of type 1 are alive through every period , while agents of type 2 are not born until a particular date is reached. In period , both types of agents are born from agents of type 1, and in successive periods all agents of a given type are born from agents of the same type.

At each date , and for each , all living agents of dynasty are treated symmetrically and take the same decision . Taking into account that, before is reached, only agents of the type 1 are born, the aggregate feasibility constraint (1) and the initial condition (4) can be represented in terms of individual decisions of the two types of agents for all t, by:

\[\begin{array}{r c l} \left(n _ {- 2} ^ {1}, n _ {- 1} ^ {1}, k _ {0} ^ {o 1}\right) & = & \left(1, \overline {{n}} _ {- 1}, \overline {{K}} _ {0}\right), \\ c _ {t} ^ {o 1} + n _ {t - 1} ^ {1} \left[ c _ {t} ^ {m 1} + b _ {t} (n _ {t} ^ {1}) + k _ {t + 1} ^ {o 1} \right] & \leq & F _ {t} (k _ {t} ^ {o 1}, n _ {t - 1} ^ {1}), \mathrm{and} \\ a _ {t} ^ {2} & = & \left(c _ {t} ^ {m 2}, c _ {t + 1} ^ {o 2}, n _ {t} ^ {2}, k _ {t + 1} ^ {o 2}\right) = (0, 0, 0, 0), \end{array}\]

whenever

\[c _ {t} ^ {o 1} + n _ {t - 1} ^ {1} \left[ c _ {t} ^ {m 1} + b _ {t} (n _ {t} ^ {1} + n _ {t} ^ {2}) + k _ {t + 1} ^ {o 1} + k _ {t + 1} ^ {o 2} \right] \leq F _ {t} (k _ {t} ^ {o 1}, n _ {t - 1} ^ {1}),\]

whenever and

\[\sum_ {j = \{1, 2 \}} N _ {t - 2} ^ {j} \left\{c _ {t} ^ {o j} + n _ {t - 1} ^ {j} \left[ c _ {t} ^ {m j} + b (n _ {t} ^ {j}) + k _ {t + 1} ^ {o j} \right] \right\} \leq F _ {t} \Big (\sum_ {j = \{1, 2 \}} N _ {t - 2} ^ {j} k _ {t} ^ {o j}, \sum_ {j = \{1, 2 \}} N _ {t - 1} ^ {j} \Big)\]

whenever

¹⁹Throughout this section, we are no longer imposing that an allocation must yield strictly positive fertility rates at every period. Therefore we abstract for the posibility that the A−dominance relation might be non-transitive.

Also, given an allocation , preferences of an agent of type born at date will be represented by a utility function

\[U _ {t - 1} ^ {j} (a) = \left\{ \begin{array}{c l} c _ {0} ^ {o j}, & \text {whenever t = -1 and j = 1 ;} \\ u \left(c _ {t} ^ {m 1}, c _ {t + 1} ^ {o 1}, n _ {t} ^ {1} + n _ {t} ^ {2}\right), & \text {whenever t = \tau and j = 1 ;} \\ u \left(c _ {t} ^ {m j}, c _ {t + 1} ^ {o j}, n _ {t} ^ {j}\right), & \text {otherwise.} \end{array} \right.\]

Note that every symmetric allocation can be identified with an asymmetric allocation such that and for all t. e e e e e e e eTaking this into account, we can adopt the following convention: for any symmetric allocation a and every non-symmetric allocation such that for all t and for some eit will be assumed that all agents of type 1 are born in the two allocations, while all agents of type 2 may be born in but not in a.

eWith this convention, analyzing whether a Millian eficient allocation is dominated (or u dominated) by an allocation is straightforward. Furthermore, this analysis yields somewhat striking results.

Proposition 15 A symmetric Millian eficient allocation cannot be

Proof. To prove Proposition 15, we simply show that for every arbitrary Millian eficient allocation , there will always exist an allocation that dominates . A particular ballocation that dominates e bcan be constructed as follows: let be such that e efor every period , and let for and for . The e b eonly components of a that remain to be specified are and e b. Choose now and in such a way that and e e eare satisfied. Observe that such pair e b e b eexists provided a is an interior allocation. With the convention given above, the eallocation bprovides all agents of type 1 (that is, all agents who were born in e e eboth a and a) with the same utility for all t, and with strictly higher utility at , which b eestablishes that a that dominates and therefore completes the proof of Proposition 15.

Proposition 16 A Millian eficient symmetric allocation might not be . In particular, a Millian eficient stationary allocation such that is u dominated by an asymmetric allocation

Proof. To prove Proposition 16, observe first that a stationary eficient allocation such that is u dominated by an allocation such that bfor all t. Thus, if a b bstationary allocation a is Millian eficient, then such allocation satisfies

bWe show next that if a satisfies , then b b bis also u dominated by some b b b b b. To prove this statement, and taking into account the convention adopted through this section, it is suficient to show that there exists an allocation such that, for one has

\[\begin{array}{r c l} U _ {t} ^ {1} (a ^ {1}) & \geq & U _ {t} (\widehat {a}), \\ n _ {t} ^ {1} & \geq & \widehat {n} _ {t}, \\ U _ {t} ^ {2} (a ^ {2}) & \geq & \underline {{u}}, \end{array}\]

and

\[n _ {t} ^ {2} \geq 0,\]

with some of the inequalities above being strict for at least one . A particular allocation that dominates can therefore be constructed as follows: let be such that efor every period t b, and let be such that for and for e b e eThe only components of a that remain to be specified are and e b. Choose now such that with (where and therefore ebut with eAnd chose b ein such a way that and b, with eObserve that such exist and e eexists given that band e bis a continuous e e b bfunction. Clearly, the allocation a is feasible and provides all agents with higher utility, which establishes that eand therefore establishes Proposition 16.

The results obtained in Proposition 15 and Proposition 16, as well as their proofs, show that opening the window to asymmetric allocations has important implications. On one side, one might argue that exploiting certain improvements or u improvements (at a certain cost in terms of symmetry) might be interesting. In the proof of Proposition 16, we constructed an allocation u dominating a Millian eficient one in which only the second type of agents born at a given date are treated asymmetrically, and infinitely many more agents are allowed to be alive and obtain utility above the threshold level This suggests that the restriction to symmetric allocations yields a too weak notion of eficiency.

On the other side, Propositions 15 and 16 show that if we drop the symmetry restriction underlying the notion of Millian eficiency, then we would need diferent criteria to compare allocations of diferent population size. Without this restriction, a symmetric allocation cannot be eficient, even if the dominance relation is transitive. Also, the set of Millian eficient stationary allocations that are not dominated by any other allocation reduces to that allocation in which all generations of agents obtain the utility threshold . These conclusions seem obviously undesirable and put some doubts on the notions of eficiency and , specially if it is also assumed that , which will imply that no Millian eficient allocation (and, possibly, no interior allocation) can be .²⁰ Furthermore, determining whether or not a society should allow for certain asymmetries is subtle. Put it in terms of intrafamiliar contracts between parents and their children like those sustaining Millian eficient allocations, allowing for that type of asymmetries would imply that parents are allowed to behave as discriminating monopolists with their own children.

To summarize, relevant critics arise to consider that the Millian notion of eficiency is the best notion available to evaluate the performance of symmetric allocations, even if we focus on a symmetric environment in which all living agents of the same generation have equal preferences. At the same time, any alternative notion of eficiency should incorporate symmetry considerations, not only because symmetry is an appealing property, but also because a notion of eficiency that does not incorporate symmetry considerations might be too strong and rule out most allocations as being ineficient. In any case, the notion of Millian eficiency seems a minimum requirement that every symmetric allocation should satisfy. To the extent that all equilibria analyzed in the literature that focus on a symmetric setting like the one analyzed in this paper are symmetric equilibria, the notion of Millian eficiency is still relevant and rules out a wide range of symmetric allocations as being ineficient, even in the absence of a precise criterium to evaluate the utility

²⁰One way of introducing symmetry considerations without restricting the set of comparable allocations to be symmetric would be by extending the notion of u−dominance in a way that the utility given to non-born agents in a given allocation depends on the utility achieved by all agents that are alive under that allocation. For example, the threshold utility level of an agent of certain characteristics who does not get to be born might be chosen to be equal to the utility achieved by the (living) agent who is worst of among all living agents with the same characteristics. However, such extension involves also challenging normative questions that exceed the scope of this paper.

obtained by non-born agents.

6 Conclusions

This paper studies the issue of Pareto eficiency in an overlapping generations setting with endogenous population. In an environment in which the set of agents is endogenous, we explore the properties of an extension of the notion of Pareto eficiency, referred to as Millian eficiency. The notion of Pareto dominance underlying the Millian notion is based exclusively on the preference profiles of those agents alive, and allows only to rank symmetric allocations (i.e., allocations in which all alive individuals of the same generation are treated equally and take the same decisions) with positive fertility rates through every period. We provide necessary conditions that every Millian eficient allocation must satisfy, and a suficient condition determining whether a given allocation satisfying these necessary conditions is Millian eficient. We show that when fertility is endogenous, the set of feasible allocations faced by agents in overlapping generations economies is non-convex, and the suficient conditions for dynamic eficiency ofered by Cass (1972) and Balasko and Shell (1980) cannot be straightforward applied. Thus, we provide an extension of Balasko and Shell’s suficient condition to non-convex settings.

With these results at hand, we adapt the Fundamental Theorems of Welfare Economics to a setting with endogenous population by characterizing every (statically) Millian eficient allocation as the equilibrium of a decentralized sequential price mechanism. Similar to the case of economies with exogenous population, every Millian eficient allocation can be decentralized by initially selecting an appropriate sequence of intergenerational transfers, and then allowing the agents to determine their consumption and investment decisions at competitive markets. Diferently from the standard, exogenous population case, an incentive scheme that links intergenerational transfers with fertility decisions is needed. More precisely, for every system of intergenerational transfers that achieves Millian eficiency, every middle-aged adult has to pay a tax (or, in some cases, receive a subsidy), and every old adult will receive a subsidy (or pay a tax) which depends linearly of the number of children she decided to have. As a particular case, we also show that the allocation corresponding to a decentralized equilibrium with no intergenerational transfers, for which there is no need to subsidize or tax children, is (statically) Millian eficient.

To summarize, the theoretical study on the notion of eficiency with endogenous populations presented in this paper has relevant implications for analyzing the role of social security programs in achieving optimal intergenerational trade. First, empirical tests of dynamic eficiency based on the P-D-K criterium might be no longer valid. Second, optimal intergenerational trade might be reached by spontaneous agreement of the agents involved. Third, if a government wishes to enforce intergenerational transfers, a mechanism linking these transfers with fertility decisions is needed.

Several extensions, such as allowing for more general forms of altruism between the agents and their descendants or introducing human capital accumulation by children, would be worth exploring. These two extensions would provide a more general setting to discuss any proposal on the role that some institutions, such as the family or the institutions comprising the welfare state, should play in achieving optimal intergenerational trade. Some authors have pointed out that, in addition to the market failure caused by dynamic ineficiencies, other types of market failures might afect intergenerational trade. For example, Becker and Murphy (1988), or Boldrin and Montes (2004) have argued that children might not have access to capital markets to finance their human capital accumulation; and Rangel (2003), has argued that the elderly cannot rely on the markets to obtain some goods. If the optimal rate of return to investment in children is afected by intergenerational transfers (as it occurs in the setting studied in this paper) any public policy that enforces intergenerational transfers on eficiency grounds should take into account the efect of intergenerational transfers on fertility choices.

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Figure 1: Stationary Dynamic eficient allocations when W is strictly quasiconcave.

Figure 2: An example when is not strictly quasiconcave of a stationary allocation e that verifies the P-D-K condition, but is not eficient .

Figure 3: A case where P-K-D and (8) criteria coincide.

Figure 4: A case where the P-D-K criterium is not able to identify eficient allocations, despite (8) condition is.

DOCUMENTOS DE TRABAJO

References

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References

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References

1. 2004-11: “Well-being Consequences of Unemployment in Europe”, Namkee Ahn, Juan Ramón García López y Juan F. Jimeno

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1. 2004-09: “Could this ever happen in Spain? Economic and policy aspects of a SARS-like episode”, José A. Herce.

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1. 2004-08: “Capital humano en España: Una estimación del nivel de estudios alcanzado”, Javier Alonso y Simón Sosvilla-Rivero.

References

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1. 2004-03: “El futuro de las pensiones en España: Perspectivas y lecciones”, J. Ignacio Conde-Ruiz y Javier Alonso.

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1. 2004-02: “Do temporary contracts increase work accidents? A microeconometric comparison between Italy and Spain”, Virginia Hernanz y Luis Toharia.

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1. 2004-01: “Job Match Quality throughout the Business Cycle in the Spanish Labour Market”, Cristina Fernández.

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1. 2003-26: “La desigualdad salarial en España. Efectos de un diseño muestral complejo”, Juan Ramón García López.

References

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1. 2003-24: “El diseño complejo de la Encuesta de Estructura Salarial 1995: Implicaciones sobre la estimación de medidas de desigualdad”, Juan Ramón García López.

TEXTOS EXPRESS

References

1. 2003-01: “12+1 Reflexiones sobre 12+1 años de Gasto Farmacéutico”, José-Luis Perona Larraz.