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**Sustaining welfare when intratemporal inequalities matter1**

**Abstract**

Sustaining the highest possible level of welfare over time has different conse-quences on regions unequally endowed with renewable resources. Consider the North being endowed with a larger stock than the South. The stock of the North is then relatively less productive at the margin. It is showed that North has to over-consume – its marginal productivity grows – while South has to under-consume – its marginal productivity rises. The steady state is reached at an equality of the marginal productivities in the two regions. The higher the intragenerational inequality aversion, the lower the highest sustainable welfare. But as long as this inequality aversion is finite, South bears a higher cost than North to sustain the global economy. The next chapter will study this feature when one no longer wants to sustain welfare, but one wants to grow. The first two chapters ignore the possibility of transfer, which is addressed in the last chapter.

**Introduction**

The growing impacts of human activity on the environment have increased concern for sustainability and call for the definition of tools to assess it. To Solow (1993), sustainability means the ability to support a standard of living for the very long-run, and requires conserving a “generalized capacity to produce economic well-being”, accounting for all components of human well-being, in-cluding the consumption of manufactured goods, the flow of services from the environment, etc. A growing body of work proposes metrics for sustainabil-ity accounting (Neumayer, 2013), among which genuine savings indicators are prominent. Genuine savings measures the evolution of the productive capacities of the economy through net investment in a comprehensive set of capital stocks. If the concerns for sustainability come from the hypothesis that society’s current decisions are not sustainable, it can hardly be held that observed, market prices can be used for sustainability accounting. Most of the genuine savings literature is based on the maximization of a welfare function, which defines a value V (X) for any economic state X (vector of capital stocks). Shadow values ¶V (X)=¶ Xi are then used to compute genuine savings as åi ¶V (X) dXi (Asheim, 2007; Das- ¶ Xi dt gupta, 2009).2 Genuine savings then measures the net investment in the capacity to produce the chosen measure of welfare.

But distributional issues happen alongside with the sustainability objective. Indeed, people living in rich countries and people living in poor countries are likely to face different challenges. This is well known in the climate change literature (Heal, 2009), but it is intimately linked with natural assets. Environ-mental problems are complex, uncertain and important, but ultimately ethical. A first step toward a better implementation of policies would be to consider di-rectly distributional issues into a sustainability objective. Do we want a general capacity of the economy (including goods and services from the environment) to produce well-being if some individuals would be more and more deprived in the future? Still, defining and assessing sustainability remains an issue in itself.

The literature on genuine savings mostly adopts discounted utility as a mea-sure of welfare. While it is the customary measure of intertemporal value in economics, discounted utility is criticized in the sustainability literature as being inequitable (Heal, 1998; Martinet, 2012). An alternative measure is the maximin value (Rawls, 1971), which is related to intergenerational equity (Solow, 1974) and defines the highest egalitarian and efficient path that could be implemented from current state in regular problems (Burmeister and Hammond, 1977). This criterion motivated Hartwick’s work on nil net investment (Hartwick, 1977), which is the backbone of genuine savings measures. For non-regular problems, though, the highest egalitarian path may not be efficient. As the maximin crite-rion does not satisfy Pareto efficiency (Asheim and Zuber, 2013), an important stream of the literature, mainly axiomatic, has focused on the definition of alter-native social welfare functions (SWFs) that encompass both economic efficiency and intergenerational equity (Chichilnisky, 1996; Alvarez-Cuadrado and Long, 2009; Asheim et al., 2012; Asheim and Zuber, 2013). This literature tries to overcome impossibility theorems stating that there is no SWF satisfying both the axiom of strong anonymity and the axiom of strong Pareto efficiency. A crite-rion relaxes either the axiom for efficiency (e.g., the maximin criterion is anony-mous but does not satisfy strong Pareto efficiency) or the axiom for equity (such as Chichilnisky’s (1996) criterion, which replaces anonymity by the axioms of non-dictatorship of the present and non-dictatorship of the future), or it has to be incomplete (such as overtaking criteria) or non-constructible.3 While this litera-ture raises interesting normative issues, most of the SWFs it produced are not as readily implementable into a sustainability accounting system as maximin.4

I see at least three interests in studying maximin for intergenerational pur-poses. First, the maximin value has a clear, positive interpretation in terms of sus-tainability, as soon as one defines sustainability as the ‘ability to sustain welfare’. This value is the highest level of welfare that can be sustained forever given the current state of the economy (Cairns and Long, 2006; Cairns, 2011, 2013; Fleur-baey, 2015a). It is my measure of ‘intergenerational equity’ herein. A genuine savings indicator can be defined for maximin, the shadow-value of a stock being its marginal contribution to the maximin value. Net investment accounted using these shadow-values, for any given dynamic path – efficient or not, and whether or not maximin is the pursued social objective, – represents the evolution over time of the highest sustainable level of utility and is interpreted as a measure of sustainability improvement or decline (Cairns and Martinet, 2014). Comput-ing net maximin investment is thus meaningful for sustainability accounting as it informs on the effect of current decisions on the ability to sustain utility that is bequeathed to future generations. Second, it guarantees a procedural equity since all generations are treated equally (finite anonymity). It seeks to maximize the welfare of the worst-off generation to the extend possible, irrespectively of its date of appearance. At the end, every generation can enjoy the maximin wel-fare, possibly constant over time. Third, if society is concerned with the effective distribution of welfare over time (consequentialist approach), it may have an in-tertemporal inequality aversion (IA). If one interprets IA through (the converse of) the elasticity of substitution (Atkinson, 1970), it may be shown that max-imin corresponds to the limiting case of infinite intertemporal IA (d’Autume and Schubert, 2008a).

The possibility of developing a sustainability accounting system based on maximin values requires defining the various capital stocks’ shadow-values. As with any other measure of welfare, the shadow-values are determined through differentiating the corresponding value function. The computation of maximin shadow-values for any actual economy, with all its various assets, consumption goods, production techniques, etc., is presently out of reach. A sensible way to proceed is to build up from the few solved maximin problems, and to try to gain a greater understanding of the economic issues involved, as it was done for discounted utility (Arrow et al., 2003). To tackle distributional considera-tions within each generation, I will consider that there are two distinct renewable resources available in two different regions, say ‘North’ and ‘South’. Besides, the general idea of genuine savings was constructed on several assets (Hartwick, 1977). I postulate an economy with two regions that do not interact together. By solving the maximin problem for this economy, I provide both some insights of the interplay between intratemporal IA and the sustainable welfare and some insights for the future development of a system of accounting based on maximin shadow-values. For the sake of simplicity, I will assume that each regional in-dicator of well-being is linear with respect to the regional consumption. This avoids inter-regional comparisons of utility, but mainly it considers regions re-sponsible for turning consumption into utility. Thus I will be interested in allo-cation of regional consumptions.

A key constituent of my examination is the question of elasticity of substi-tution (EOS) between consumption of North and consumption of South. This refers both to IA and substitutability issues. The ‘welfarist’ approach allows for dealing with two famous limiting cases, as well as all intermediary cases. Indeed, utilitarianism is concerned with the global welfare irrespectively of its inter-individual distribution. It is characterized by a nil IA (infinite EOS). At the opposite, intragenerational maximin is concerned with the worst-off individual. It corresponds to an infinite intratemporal IA (nil EOS). Interestingly, this echoes concerns with substitutability between different stocks, in North and South here. Neumayer (2013) stresses that substitutability in production as well as in welfare plays a central role in the study of sustainability. The influence of substitutability in production on the maximin solution has been emphasized since the work of Solow (1974) and Dasgupta and Heal (1979), who studied interactions between sectors in the form of a sector extracting a non-renewable resource used as an input to a manufacturing sector. Some authors (e.g. Asako, 1980; Stollery, 1998; d’Autume and Schubert, 2008a; d’Autume et al., 2010) study maximin problems with two substitutes in welfare (utility), one of which is a decision variable (con-sumption) and the other a state variable (the ambient temperature or the stock of a non-renewable resource). Substitutability of consumption goods in welfare has received less scrutiny but is as important a question for sustainability as substi-tutability in production.5

Keeping the economy at a steady state by maintaining current capital stocks is a proposal that appeals to some proponents of sustainability (Daly, 1974; Neu-mayer, 2013). By solving the maximin problem for our economy, it is shown in Section 1.2 that, whenever the IA is finite (but non-zero), if regions are dif-ferently productive at the margin, it is possible to sustain welfare at a higher level than at that regional steady state. There are a higher consumption in the marginally less productive region (North) and a lower consumption in the more productive one (South). From a social welfare point of view, the depletion of the less productive stock is compensated for by investment in the more productive one, which echoes the Hartwick’s nil net investment rule (Hartwick, 1977; Dixit et al., 1980; Cairns and Long, 2006). This investment pattern is driven by the shadow-values of the stocks, i.e., the sustainability accounting prices. What is maintained is a general capacity to sustain welfare, not levels of particular natural assets. This calls for transfers from North to South as discussed later on.

In single-sector models such as the Solow (1956) growth model with capital depreciation or the simple fishery, stocks that are beyond the golden-rule level or the maximum sustainable yield have negative marginal products and are redun-dant for maximin value (Solow, 1974; Asako, 1980). In such a non-regular case, an egalitarian path is not efficient (Burmeister and Hammond, 1977), and the maximin value and shadow-values are of little information. It is thus important to identify the conditions under which such non-regularity occurs. In our two-sector model, there is no such inefficiency when abundant stocks can be used in the investment pattern (in North) to build up a scarce resource (in South), and so have a positive sustainability accounting value. Stock redundancy, which is associated with nil sustainability accounting prices, arises only if all technologies have a single productivity peak, and is thus less likely to occur in a multi-region model with substitutability. Positive shadow-values directly indicate scarcity.

It is shown is the Section 1.3 that substitution/investment pattern is influenced by the degree of intragenerational IA between regional consumption goods in a subtle way, in interplay with the relative ‘productivity’ of the natural stocks. Moreover, in the limiting case of infinite IA, the region (South) with the less abundant resource limits sustainability and the more abundant one is redundant (North). Only the former has a positive accounting price for intergenerational equity.

The consequences of these results for sustainability accounting with max-imin shadow-values are discussed in Section 1.4. In particular, two conditions for current decisions to improve the level of welfare that can be sustained are de-termined. First, current welfare has to be lower than the maximin value. Second, the resource thus freed-up must be invested in order to get a positive maximin net investment. Llavador et al. (2011) stressed that the year 2000 consumption in the USA was lower than the sustainable, maximin value. Such a lower welfare can be consistent with long-run growth as long as both investment decisions re-sult in an increase of the maximin value and proper transfers toward developing countries are implemented.

Conclusions and prospects for future research are given in Section 1.5. Ad-ditional mathematical details and analyses (for special cases, in particular) are provided in the Appendix.

### Two regions, two reproducible assets

In this section, the maximin solution in a two-region model is characterized: each region, North and South, having access to a distinct renewable resource. A social planner derives an instantaneous social welfare function (SWF) W from utilities of the regions. For the sake of simplicity, utilities are assumed to be linear: u(ci) = ci; i = N; S. This avoids utility comparisons and it considers re-gion responsible for turning consumption into utility. I start with neoclassical assumptions on production. Then single-peaked technologies (SPT) are considered, introducing a source of stock redundancy which happens generally with renewable resources (e.g. a logistic growth).

**A neoclassical benchmark**

Consider an economy with two reproducible assets, XN and XS, produced by separate regions, North and South, according to technologies Fi(Xi), which depend only on the stock Xi and are assumed to be twice continuously differentiable, strictly increasing (Fi0 > 0)6 and strictly concave (Fi00 < 0).7 Production is either consumed (ci) or ‘invested’ X˙i and capital dynamics are ˙ dXi(t) (1.1) Xi(t) dt = Fi(Xi(t)) ci(t) ; i = N; S :

The economy is composed of infinitely many generations of identical con-sumers, each living for an instant in continuous time. Consider an ordinal so-cial ordering over the two consumptions, represented by a twice-differentiable, strictly quasi-concave and symmetric SWF W (cN ; cS), such that both goods have a positive marginal social welfare and are socially ‘essential’.8

The maximin value m of a state (XN ; XS) is the highest level of welfare that can be sustained forever from that state:

m(X1; X2) = max w ; (1.2)

w;cN ( );cS( )

s.t. (XN (0); XS(0)) = (XN ; XS) ;

˙ (t)) ci(t); i = N; S; and

Xi(t) = Fi(Xi

W (cN (t); cS(t)) w for all t 0 : (1.3)

Herein, the term value refers to maximin value. Below, I omit the time argument in the expressions where no confusion is possible.

Differentiation of the maximin value with respect to time yields the net max-imin investment (Cairns and Martinet, 2014, Lemma 1):

dm(XN ; XS) ¶ m(XN ; XS) ˙ ¶ m(XN ; XS) ˙ (1.4) = XN + XS : dt ¶ XN ¶ XS

The links between the maximin problem and net investment have been studied since the work of Hartwick (1977), with recent contributions by Doyen and Mar-tinet (2012) and Fleurbaey (2015a). The links between net maximin investment and sustainability accounting are studied in Section 1.4.

Before solving the maximin problem for this economy, let us establish the following lemmata.

Lemma 1 (Stationary fallback). For any state (XN ; XS), the maximin value is at least equal to the welfare derived from consumption at the corresponding steady state:

m(XN ; XS) W (FN (XN ); FS(XS)) :

The dynamic path ˙ = 0 driven by decisions = ( ) is

Proof of Lemma 1. Xi ci Fi Xi feasible and yields the constant utility W (FN (XN ); FS(XS)). This provides a lower bound for the maximin value.

Lemma 1 relies on the fact that consuming the whole production, keeping the economy in a steady state, makes it possible to sustain W (FN (XN ); FS(XS)). A dynamic path may, however, yield a higher sustainable utility.

Lemma 2 (Dynamic maximin path). If the maximin value of a state (XN ; XS) is greater than the welfare derived at the steady state, i.e., if m(XN ; XS) > W (FN (XN ); FS(XS)), then, along the maximin path (i) the consump-tion of at least one good is greater than the production of the corresponding stock and (ii) that stock decreases.

Proof of Lemma 2. This is a direct result from Lemma 1 and the dynamics. The existence of such a dynamic path means that keeping the economy at a steady state (Daly, 1974) is not the only sustainable option. Solving the maximin problem (1.2) may provide a superior path. To do so, I follow the direct approach to maximin proposed by Cairns and Long (2006). Taking the sustained utility level w as a control parameter and denoting the costate variables of the stocks by mi, the Hamiltonian writes: H (X; c; m) = mN XN + mSXS = mN (FN (XN ) cN ) + mS (FS(XS) cS) : (1.5)

Denoting the multiplier associated with the constraint (1.3) by w, the Lagrangian is L (X; c; m; w; w) = H (X; c; m) + w (W (cN ; cS) w) : (1.6)

In an interesting problem, both initial stocks are strictly positive, i.e., Xi(0) > 0 for i = N; S (otherwise, one is back to the single sector problem). Under the condition that both goods are essential to consumption, and given Lemma 1, one can say that consumption of both goods is positive (ci > 0; i = N; S) at any time along a maximin path. The necessary conditions are, for i = N; S, and for any time t:

Whenever equality is efficient, the maximin path corresponds to an egalitar-ian and efficient path, with welfare equal to the constant maximin value. The solution is said to be regular (Burmeister and Hammond, 1977) and corresponds to a (strong) Pareto allocation of utility among generations. The variable w mea-sures the shadow cost of the equity constraint, i.e., how much would be gained in value if the constraint was locally relaxed. This is the opportunity cost of meeting the constraint at the current state.9

Cairns and Long (2006, Proposition 1) show that along a maximin path, the mi are the shadow-values of each stock at current state, i.e., mi = ¶ m(XN ;XS) , and that the Hamiltonian, which thus represents net investment at maximin shadow-values (eq. 1.4), is nil: H (X; c; m) = mN XN + mSXS = m˙ Eq. (1.14) is related to Hartwick’s rule (Hartwick, 1977; Dixit et al., 1980; Witha-gen and Asheim, 1998). The maximin value is constant over time and, as welfare is equal to the maximin value (W (cN ; cS) = m(XN ; XS)), so is welfare.

The optimality conditions above can be given economic meanings in a regu-lar problem, when (mN ; mS; w) 6= (0; 0; 0).10

From eq. (1.7), the shadow-value of stock Xi is equal to the marginal utility of consumption ci weighted by the shadow-value of equity mi = wWci ; i = N; S : (1.15)

As long as w > 0, the relative shadow-value is equal to the marginal social rate of substitution in consumption: mN = WcN : (1.16)

From eq. (1.8), m˙i = miFi0(Xi), so that each shadow-value decreases at a rate equal to the current marginal product of the corresponding stock: mi = F 0 (Xi) ; i = N; S : (1.17)

This depreciation rate is the cost of postponing an investment over a short period of time (see Dorfman, 1969, p. 821). The lower a stock, the higher its marginal product and the more costly in terms of maximin value it is to postpone invest-ment in the stock.

**Table of contents :**

**General introduction **

0.1 Context

0.2 Intragenerational equity

0.2.1 Historical background

0.2.2 Axiomatic approach

0.2.3 Selected approach

0.3 Intergenerational equity

0.3.1 Historical background

0.3.2 Axiomatic approach

0.3.3 Intertemporal criteria

0.3.4 Selected approach

0.4 Linking the two dimensions

0.5 Research questions

0.6 Outline and results

**1 Sustaining welfare when intratemporal inequalities matter **

1.1 Introduction

1.2 Two regions, two reproducible assets

1.2.1 A neoclassical benchmark

1.2.2 Single-peakedness: A source of stock redundancy

1.3 Inequality aversion and sustainability

1.4 Accounting for changes in sustainability

1.5 Conclusion

A.1 Illustrations for the limiting cases

A.1.1 Illustration: Utilitarianism

A.1.2 Illustration: Intragenerational maximin

A.2 Sustainability improvement: Graphical illustrations for the limiting cases

A.3 Stability of the steady state

A.4 Utilitarianism: Mathematical details and proofs

A.5 An example: Utilitarianism with AK-technology and a renewable resource

A.5.1 Case 1

A.5.2 Case 2

**2 Distributional considerations during growth toward the golden rule **

2.1 Introduction

2.2 Maximizing intergenerational welfare

2.2.1 Framework

2.2.2 Two regions considered separately

2.2.3 Two regions considered collectively

2.2.4 Graphical representation

2.3 Accounting considerations

2.4 Intra and intergenerational inequity

2.5 Discussion: Equity, inequality aversion and discount rate

2.6 Conclusion

A.1 Stability of the steady state

**3 Transfers of resources for a sustainable development **

3.1 Introduction

3.2 Two regions, one harvestable resource

3.2.1 Framework

3.2.2 Utility possibility frontier with lump-sum transfer

3.2.3 Utility possibility frontier with tax

3.2.4 Comparison of the two frontiers

3.2.5 Welfare analysis

3.3 Intra and intergenerational considerations

3.3.1 Inequality aversion and the current consumption

3.3.2 Evolution of the resource and the possibilities for futures welfares

3.3.3 Interactions between the two dimensions

3.4 Conclusion

A.1 Special cases of CES functions

A.2 Illustration: Sensitivity to the utility of the South

**General conclusion **

3.2.1 Summary

3.2.2 Limits and prospects

**Bibliography**