David Guerreiro
email: david.guerreiro@u-paris10.fr

Marc Joëts
email: marc.joets@u-paris10.fr

Valérie Mignon
email: valerie.mignon@u-paris10.fr.

Is price dynamics homogeneous across Eurozone countries? *

Keywords: JEL Classification: C23, E31, F15, F41 price convergence, Eurozone, panel unit root tests, half-life

The aim of this paper is to investigate whether price dynamics is homogeneous across the Eurozone countries. Relying on monthly data over the January 1970-July 2011 period, we test for the absolute purchasing power parity (PPP) hypothesis through the implementation of second-generation panel unit root and cointegration tests. Our results

show that price dynamics are heterogeneous depending on both the time period and the considered group of countries. More specifically, while PPP is validated for the core EMU countries, this hypothesis does not hold for Northern peripheral economies. Turning to the Southern countries, PPP is observed only before the launch of the euro.

Introduction

Since 2009, EMU is experiencing serious economic troubles that now challenge its stability, and, to a certain extent, even its existence. Triggered by high debt ratios relative to GDP, high deficits, and low growth expectations, a sovereign debt crisis has arisen in peripheral countries. Greece, Ireland and Portugal have been forced to ask the Troïka1 for financial assistance, while Spain, even if it has succeeded to avoid such assistance, has experienced a large increase in its debt financing costs. The crisis has also reached the core of EMU, as evidenced by the large increase of spreads between Italian and German bonds, and, to a lesser extent, between French and German bonds.

If the roots of such a crisis are obviously multiple, one of the main reason would be the possible heterogeneity of price dynamics among members. In order to constitute a viable currency area, the founding members of Eurozone have developed, since the beginning of the EMU, a number of rules to ensure economic convergence, as stated in the Maastricht Treaty (1992) completed by the Pact of Stability and Growth (1997). One of the crucial rules is surely price stability: members' inflation rates should not exceed by at most 1.5 points the mean of the three lowest inflation rates. Although this is intended to render the "one-size-fits-all" monetary policy efficient, it also has consequences on the price dynamics that may differ with the reality of the EMU.

Inside a monetary union, implementing the convergence of inflation rates is intended to avoid asymmetric disequilibria coming from a loss of price competitiveness. However, it also supposes strong economic homogeneity regarding per capita income and productivity levels. More importantly, inflation rates convergence in a monetary union also implies price levels to be close to those determined by the absolute purchasing power parity (PPP) hypothesis.

Such a constraint may be viewed as very strong given the economic differences between the core of EMU and the periphery. Balassa-Samuelson phenomena inducing price catching-up are expected in peripheral countries, leading to heterogeneous convergence speeds depending on the group of countries-the extreme case being non convergence of price levels-and challenging absolute (and hence relative) PPP. In that case, common monetary policy may be inadequate because of its pro-cyclicality. On one hand, it could be too restrictive for advanced economies, and lead to deflationary pressures. On the other hand, it could be too incentive for countries experiencing a catching-up: because of negative or low real interest rates, debt bubbles may appear, conducting to worsen internal imbalances (excess leverage, asset price bubbles), and external deficits. Testing for absolute price convergence would allow us to investigate whether these processes may have materialized after the launch of the euro, partly conducting to the sovereign debt crisis.

Only a few papers deal with price convergence inside the EMU. Articles generally focus on the reasons explaining heterogeneity in inflation rates (see for example [START_REF] Engel | Deviation from Purchasing Power Parity: causes and welfare consequences[END_REF], [START_REF] Honohan | Divergent inflation rates in EMU[END_REF], or Arnold and Verhoef ( 2004)), but very few investigate price dynamics empirically (i.e. validation of absolute PPP), mainly because of the lack of available data. The differences between the existing studies come from the datasets used. Some rely on microdata- [START_REF] Engel | European product market integration after the euro[END_REF], [START_REF] Crucini | Understanding European real exchange rates[END_REF] and [START_REF] Rogers | Monetary union, price level convergence and inflation: how close is Europe to the USA?[END_REF]-while others use aggregate data-Camarero et al. (2000), Allington et al. (2005), or [START_REF] Faber | A short history of price level convergence in theory in Europe[END_REF].

Among the studies using microdata panels, [START_REF] Engel | European product market integration after the euro[END_REF] and [START_REF] Rogers | Monetary union, price level convergence and inflation: how close is Europe to the USA?[END_REF] consider data from the Economic Intelligence Unit and pay attention to the price of "standard" goods measured in 18 cities belonging to the Eurozone. They find a reduction of price dispersion during the implementation of the Single Act (1986), but the introduction of the euro seems to have had little effect. [START_REF] Crucini | Understanding European real exchange rates[END_REF] test the validity of LOOP (Law Of One Price) using Eurostat data for four different years (1975, 1980, 1985 and 1990). They show that LOOP is violated only for a little numbers of products, tending to validate absolute PPP.

Studies using aggregate data provide results that go in the same direction. Camarero et al. (2000) evidence a catching-up effect for peripheral countries (Spain, Italy, and UK) relative to Germany, by implementing time series unit root tests with structural breaks on CPI differentials.2 Allington et al. (2005), employing Comparative Price Levels (CPLs) provided by Eurostat, put forward a significant effect of the euro on price dispersion. By recalculating CPIs that account for price level differentials between countries, [START_REF] Faber | A short history of price level convergence in theory in Europe[END_REF] uphold the diminution in price dispersion across Eurozone countries. Finally, [START_REF] Guerreiro | On price convergence in Eurozone[END_REF] highlight some heterogeneity across the Eurozone members in terms of price convergence speed, that can be explained by the evolution of price competitiveness, rigidities on the labor market and technological specialization patterns.

Our aim in this paper is to contribute to this literature that is still in its infancy by investigating the homogeneity of price dynamics inside EMU over the January 1970 to July 2011 period. We show that to respect price stability-and hence keep the monetary union viable-one of the conditions lies in the achievement of absolute PPP. Therefore, to investigate price convergence we rely on unit root and cointegration tests. We go further than the usual time series and panel tests by accounting for both heterogeneity and cross-section dependence through the implementation of second-generation panel unit root tests. We also assess speeds of price convergence using the Pooled Mean Group estimator (PMG) of [START_REF] Pesaran | Pooled mean group estimation of dynamic heterogeneous panels[END_REF].

The rest of the paper is organized as follows. Section 2 presents the theoretical framework and the data. Section 3 displays the results of unit root tests and PMG estimates, while Section 4 discusses our findings. Section 5 concludes the article.

2 Theoretical backgrounds and data

Theoretical backgrounds

In a two-countries model, absolute PPP is defined by:

e t = p t -p * t (1)
where e t is the log of the spot exchange rate, and p t and p * t respectively denote the log of the domestic and foreign price levels. Relative PPP holds when:

∆e t = ∆p t -∆p * t (2)
with ∆e t denoting the variation of the log of the exchange rate, ∆p t and ∆p * t the inflation rates of domestic and foreign country. If we assume now that these countries form a monetary union with a single currency, e t = 0, and it comes for absolute PPP:

p t -p * t = 0 (3)
while relative PPP becomes:

∆p t -∆p * t = 0 (4)
Plugging (3) and (4), we get :

p t -p * t = ∆p t -∆p * t (5)
Consequently, in a monetary union, absolute and relative PPP are equivalent. In other words, convergence in inflation rates is conditioned by the fact that price levels are the same between countries. If price levels differ, the viability of the monetary area is challenged on the long run. Countries that have lower price levels are expected to catch-up their delay (Balassa-Samuelson effect), leading to higher inflation rates which goes against relative PPP inside a currency union. In a highly integrated currency union where trade barriers have disappeared, and assuming that the adjustment towards absolute PPP is made only through the goods market, a violation of absolute PPP "signals an imbalance in the goods markets which [...] is likely to result in trade deficits" [START_REF] Juselius | Wage, price, and unemployment dynamics in the convergence to PPP in the Euro Area[END_REF]). Capital inflows (thanks to borrowing) are necessary to recover the equilibrium of the balance of payments. However, these massive inflows pressure upward nominal interest rates of low price countries leading to an increase of the spreads relative to high price level countries. Accumulation of deficits also leads to an increase of the debt level which in turns raises the risk premium that traduces the default risk.

The combined effect of these two mechanisms-that reveals the mistrust to currency union viability-entails prohibitive financing rates, conducting to a sovereign debt crisis that spills over the union. This can be seen as a pure asymmetric shock as that presented by [START_REF] Mundell | A theory of optimum currency areas[END_REF] in the seminal paper on Optimum Currency Areas (OCA), that challenges the existence of the currency union. has been chosen as the basis year because it corresponds to the year of the last ICP survey realized by the World Bank.5 From Equation ( 7), we thus obtain 12 series of price levels6 that can be used to test for price convergence (see Figure 1 in Appendix). Given the importance of Germany in the Eurozone, we retain this country as the benchmark, and investigate PPP between each domestic price level series and the German one.7

Data

3 Tests and estimation results

Time series analysis

To assess PPP between each Eurozone member and Germany, we rely on unit root and cointegration techniques by estimating an ADF-type equation:

8 ∆x t = k + φx t-1 + p i=1 φ i ∆x t-i + t (8)
where x t denotes the differential between domestic (P t ) and German (P * t ) prices (in logs):

x t = ln P t -ln P * t (9)
When the null hypothesis φ = 0 in Equation ( 8) is rejected, the price differential is nonstationary, meaning that PPP does not hold. In the case where the null is rejected, it is possible to calculate the half-life of deviations (-ln(2)/ ln(1 + φ)) which provides an indication regarding the speed of price mean reversion.

As evidenced by the typology of Balassa (1961), monetary union is supposed to be the final stage of an economic integration process. Integration tends to make disappear the "boarder effects", that are the measures intended to discriminate foreign goods and services relative to domestic ones. There are several kinds of discriminations: tariff barriers (as custom duties), non-tariff (quotas), or institutional (money). All these obstacles are found to hamper PPP. 8 Previous to the application of cointegration tests, a battery of unit root tests have been applied showing that all price series are integrated of order one. The detailed results are available upon request from the authors. Note that Equation (8) does not include a deterministic trend, the latter being non significant. the Single Act. The latter defines the beginning of the second period, spanning from July 1987 to December 1998. The last period starts with the introduction of the euro in January 1999 and ends in July 2011. Those periods, which are distinguished here according to the integration process, are also characterized by different exchange rate regimes: a "smooth" European Monetary System (EMS) allowing some parity adjusments for the first period; strengthening of fixed exchange rates within the EMS, with decreasing possibilities of parity adjustments and the start of a nominal convergence process to meet Maastricht criteria and to enter in the monetary union for the second period; introduction of the monetary union in the third period.

Results of cointegration tests are presented in Table 1 which displays the p-values relating to the ADF-type test 9 and the half-life of deviations (in years). With the exception of some few special cases, 10 PPP does not hold since the tests fail to reject the null hypothesis of no cointegration between domestic and German prices. These poor results and the absence of obvious convergence may come from the well-known low power of time series unit root tests against the stationary alternative in small samples. To overcome this limitation, we rely on a panel framework. Adding the individual dimension to the analysis, the use of panel data increases the power of unit root tests by raising the number of observations. Furthermore, this approach allows us to consider both heterogeneity and cross dependence between Eurozone countries.

Panel analysis

Historically, European countries have tied links for a long time, especially since the beginning of economic integration initiated after World War II. In this context of integration, European countries have shared common specificities-one of the tremendous example being the euro, and all the domains of monetary policy. However, despite this apparent homogeneity, some heterogeneity may persist due to distinct economic characteristics across members: European construction has been done by steps, and it widens gradually incorporating new countries often considered to be less integrated to Eurozone than the founding members. Therefore, some countries known as peripheral members (like Portugal, Spain, or Greece) exhibit lower levels of GDP per capita, prices, or productivity. Assuming that Germany is at the center of our Eurozone conceptualization, we can, as Braudel (1985), divide EMU into three sub-samples of countries:11

• the Core EMU, composed by Austria, Belgium, France, Italy, Luxembourg and the Netherlands;

• the Northern periphery, including Ireland and Finland;

• the Southern periphery made by Greece, Portugal, and Spain.

We assess potential price convergence dynamics across each sub-group using panel data unit root tests. To overcome the well-known problem of cross-sectional independence hypothesis among panel members (i.e. among the Eurozone countries), we apply second-generation panel unit root tests that relax this restrictive assumption required by first-generation tests.12 ,13

The Pesaran (2007) CIPS and the Choi ( 2002) tests are performed and the corresponding results are reported in Table 2, both tests considering the unit root as the null hypothesis.14

The former test is based on Dickey-Fuller-type regressions augmented with the cross-section averages of lagged levels and first differences of the individual series. The later relies on an error-components panel model and removes the cross-section dependence by eliminating (i)

individual effects using the [START_REF] Elliott | Efficient tests for an autoregressive unit root[END_REF] methodology (ERS), and (ii) the time trend effect by centering on the individual mean.

Results are relatively mixed and dependent on the test considered. The CIPS and Choi's tests frequently lead to different results. This difference may come from the way cross-sectional dependence is accounted for. Both tests are based on the one common factor approach. However, while Choi's test uses the orthogonalization procedure to deal with the cross-dependence problem, the CIPS test uses augmented cross-section average and therefore does not eliminate individual and time effects. In addition, while both tests consider one common factor, only the Choi's test allows for heterogeneous specification regarding the sensitivity of the factor, which is particularly relevant in the Eurozone macroeconomic context. Paying thus a particular attention to the Choi's approach, PPP relative to Germany is validated for both the whole sample and the core EMU in the three sub-periods. PPP also holds for Southern periphery during the first two periods, but not after the launch of the euro. Regarding Northern periphery, there are strong evidences that PPP is violated for each sub-period. On the whole, our results put forward that price dynamics is heterogeneous between EMU members. To complement these findings, we estimate the half-life of deviations from PPP for sub-groups as well as individual countries. To this end, we use the PMG estimator developed by [START_REF] Pesaran | Pooled mean group estimation of dynamic heterogeneous panels[END_REF]. This procedure consists in estimating a panel error correction model where homogeneity accross members is imposed for the long-run relationship,15 while heterogeneity is allowed in the short-run dynamics and error correction mechanisms. It combines two well-known procedures used in panel data: (i) the totally heterogeneous "mean group estimate", that consists in estimating separately N individuals regressions and averaging the group specific coefficients; and (ii) the homogeneous "pooled estimator", in which only the intercept is authorized to differ across Eurozone members.

Tables 3 and4 report the PMG estimates of the error-correction term and the corresponding half lives. As shown in Table 3, the period corresponding to the implementation of the Single Act is the one for which half lives are the smallest for the whole sample (about 6 months)

and for the Core EMU (about 4 months). This is in line with the literature arguing that the Single Act has allowed to fully remove the remaining trade barriers [START_REF] Engel | European product market integration after the euro[END_REF]). This may also be linked to the requirements of nominal convergence due to the entry into the monetary union. The introduction of the euro, that was expected to accentuate this mechanism by improving market information, lowering trade costs and reducing uncertainty, seems to have had little effects, since half lives have increased (12 months for the entire sample, 10 for the Core EMU)-a finding that may be related to weaker ajustment mechanisms within the monetary union. Turning to Southern periphery, half lives are very small in the first period (about 6 months), but increase in the second (10 months). This increase may be obviously linked to the EMS crisis and, especially, to the huge nominal devaluation encountered by the Spanish and Portuguese currencies following the enlargement of the fluctuation margins within the monetary union.

Regarding individual countries (Table 4), heterogeneity across members is even more marked.

Inside the Core, and during the first period, France exhibits the lowest half life, about 5 months, while it is about 34 months in Italy and 115 in Austria. During the second period, the differences tend to diminish, but the launch of the euro triggered a raise of gaps. One of the worrying result is the half life of Belgium in the third period, being about 68 months.

Such a value may indicate that this country is running out the cointegration path. Regarding the Southern group, departures from PPP have increased between the first and the second period, but they are relatively homogeneous between countries.

Analysis and scope of the results

Our results put forward that price dynamics is not homogeneous (i) in time, and (ii) between European countries. Here, we aim at identifying the reasons of this heterogeneity across members, a fact that is highly relevant given that heterogeneity in price dynamics is at the roots of the sovereign debt crisis that EMU is experiencing.

Heterogeneity of price dynamics as a consequence of structural economic heterogeneity

One of the reasons explaining the differences in price dynamics is the Balassa-Samuelson catching-up effect, that entails an elevation of the aggregate price levels in the low-productivity level countries. It can be considered as an asymmetric shock since the high-level productivity countries are expected to experience price stability. One of the proxies often used to model this phenomenon is the relative GDP PPP per capita.

Table 5 reports GDP PPP per capita relative to Germany. The 2007 value is shown to assess the effects of "subprimes" and hence "sovereign debt" crises. Regarding Southern Europe, the situations are somewhat mixed. Portugal has markedly reduced the gap relative to Germany before 1999, however it is not totally fulfilled and differences are still consequent. Moreover, the reduction is definitely stopped after 1999 with the launch of the euro. Instead, Greece seems to have experienced a catching-up after the implementation of the single currency. As

Portugal, the departure is not eliminated. Finally, Spain appears to have converged to German GDP per capita all over the period. Balassa-Samuelson catching up is then a satisfactory explanation for the PPP violation for Greece and Spain, but not for Portugal. Regarding

Northern periphery, the Balassa-Samuelson catching up is at play in Ireland and Finland for the whole period. Turning to the Core, interesting insights stand out. GDPs per capita relative to Germany have been relatively stable between periods (unless Luxembourg, that can be considered as an outlier), providing a general background for the results found in Section 3.2. The elevated half-lives obtained for Italy during all the periods can be explained by the size of the GDP gap (that is relatively higher than for the other countries belonging to the Core) relative to Germany.

Nonetheless, Balassa-Samuelson catching up is not the unique reason of asymmetric shock appearance. The nominal exchange rate level relative to the rest of the world can also constitute a factor that can lead to divergences in price dynamics. In other words, can EMU members compete with low wage countries with a strong currency if they are specialized in low technology products? Since the competitiveness of these products is highly sensitive to costs, members specialized in low technology have to pressure downward their wages, and also their price to recover competitiveness relative to low wage countries. Members that are specialized in high technology do not experience these constraints since their products are less sensitive to costs. Two different price dynamics are then at play: one pressured downward, the other unconstrained. Since the wages are not likely to decrease, members specialized in low technology will face a dramatic loss of competitiveness if they do not go up market.

We try to control for that effect by constructing a simple index of technological specialization on the basis of the revealed comparative advantages for the CITI manufacturing indicator (see Appendix for the construction of the index). Table 6 reports the technological specialization for each country during the periods 1987-1998 and 1999-2009. Four categories of countries can be distinguished:

• Our benchmark country, Germany is specialized in industries with high technological contents: its products are innovative, or positioned on the high quality segment, which allows it to discriminate its prices.

• Austria, Finland, France, and the Netherlands own at equal shares high and medium technology industries. They have a lower market power than Germany, even if it remains substantial.

• Belgium, Ireland, Italy and Spain are mainly specialized in medium-technology sectors, and have very few high-tech industries. Moreover, low-technology sectors represent an important part of specialization in Spain and Italy.

• Greece and Portugal are mostly specialized in low-technology industries, facing a strong competition in prices.

The heterogeneity of specialization supplemented by rigidities in wage decrease, may explain the non-convergence of Southern countries, especially Portugal, and also the elevated half-lives of Belgium and Italy: without the possibility of decreasing wages (and hence prices), a strong currency induces a loss of competitiveness that translates into imbalances in goods markets, and finally a violation of PPP.

Scope of the results

To (1976)(1977)(1978)(1979)(1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995). Such results would mean that EMU is more integrated than the US, which is unlikely. The more plausible explanation lies in the data used as well as the econometric techniques that have been implemented. Cecchetti et al. (2002) consider price indexes instead of price levels, which is not appropriate for testing absolute PPP. Moreover, first generation panel unit root tests may have been too restrictive to assess price convergence since they assume homogeneity between members, an hypothesis that is not verified as seen in the previous section.

The recent literature on OCA has mainly devoted attention to cyclical rather than structural homogeneity of the countries considered. On one hand, an empirical strand tried to assess the correlation of business cycles and the symmetry of shocks through VARs (see Bayoumi andEichengreen (1992, 1996) among others). On the other hand, a theoretical strand focused on the means of rendering business cycles and shocks more symmetric (the endogenous criteria literature, pioneered by Frankel and Rose (1998)). Our approach revives the pioneering phase of OCA literature [START_REF] Mundell | A theory of optimum currency areas[END_REF][START_REF] Mckinnon | Optimum currency areas[END_REF] and [START_REF] Kenen | The theory of optimum currency areas: an eclectic view[END_REF]) that emphasized on the structural similarities of currency union members, and stresses that these criteria may be the most important to achieved the optimality, or at least the viability, of a monetary union. Following [START_REF] Juselius | Do purchasing power parity and uncovered interest rate parity hold in the long run? An example of likelihood inference in a multivariate time-series model[END_REF], linking PPP (goods market) and uncovered interest parity (capital market) conditions, it is possible to understand through price dynamics why structural heterogeneity among members of a currency union is a problem. In the absence of trade barriers as well as exchange rate adjustments, a violation of PPP (such a Balassa-Samuelson catching-up, but not only) leads immediately to goods market imbalances (trade deficits) that have to be compensated by capital inflows. This creates asymmetric debt emissions between members that results in fine (through long interest rates and default probability differentials) in sovereign debt crisis from part of the members. This mechanism seems to fit the troubles EMU is experiencing.

Conclusion

This paper aims at investigating whether price dynamics is homogeneous across the Eurozone countries. This question if of particular interest in the current debt crisis context, since heterogeneity in price dynamics may be one of the causes of the turmoil. Relying on monthly data over the January 1970-July 2011 period, we test for the absolute purchasing power parity (PPP) hypothesis through the implementation of second-generation panel unit root and cointegration tests. Our results show that price dynamics is heterogeneous depending on both the time period and the considered group of countries. More specifically, while PPP is validated for the core EMU countries, this hypothesis does not hold for Northern peripheral countries such as Ireland and Finland which experienced a Balassa-Samuelson effect. Turning to the Southern countries, namely Greece, Portugal and Spain, PPP is observed only before the launch of the euro. This violation of the PPP on the recent period may be explained by a catching-up phenomenon, as well as the countries' technological specialization pattern.

Our findings of heterogeneous price dynamics may be viewed as one the possible causes of the current debt crisis and contribute to the debate relative to the viability of the monetary union. Note that index 23 has a high technology industry, that of nuclear, in which only France is specialized. Accordingly, we consider that for France index 23 testifies a high technology specialization. In Table 8, we report for the two sub-periods the sectors in which countries are specialized.

Bibliography

Allington, N.F.B., Kattuman, P.A. and F.A. Waldmann (2005), "One Market, One Money, One Price?", International Journal of Central Banking 1(3), 73-115.

Arnold, I.J. M and B.A. Verhoef (2004), "External causes of euro zone inflation differentials.

A re-examination of the evidence", Intereconomics 39, 254-263. Balassa, B. (1961), "Towards a theory of economic integration", Kyklos 14, 1-17.

  Consequently, we consider three sub-periods that are in line with the different steps of EMU construction. The first period, starting in January 1970, sticks with the Common Market (established by the Rome Treaty in 1953) and ends in June 1987 with the implementation of benchmark price level series.

  our best knowledge, with the exception ofCecchetti et al. (2002), no study exists on the PPP deviation inside a monetary area.Cecchetti et al. (2002) have tested price index convergence among US cities using first generation panel unit root tests. Comparing our results with theirs may be interesting because the US is a long-lived monetary area, with strong political links and where economic integration is entire. AlthoughCecchetti et al. (2002) find, as we do, price convergence between US cities (i.e. validation of the PPP hypothesis), their half-lives are far higher than ours. They are comprised between eight and nine years for their whole sample (19 cities on the 1918 to 1995 period), and about eight years for the more recent period

  P i,t stands for the PPP conversion factor for private final consumption of country i relative to the United States at time t, expressed in euros per US dollar, and N ER i,t is the euro/dollar exchange rate at time t. Turning to data availability, CPLs are computed by Eurostat and OECD for each European country annually only since 1995. It is however possible to recover observations previous to 1995 using the price evolution relative to the

	CP L i,t =	P P P i,t N ER i,t	× 100	(6)
	where P P
	To test for absolute PPP, price levels of Eurozone countries are needed. CPIs are unusable
	since they do not account for price level differentials between countries: they only depict
	the evolution of prices. To overcome this issue, we rely on the International Comparison
	Program (ICP) carried by the World Bank that aims at providing comparable international
	prices. Based on this framework, Eurostat and OECD have computed Comparative Price

Level (CPL) series for each European country. These CPLs are defined by the OECD as the ratio between purchasing power parity conversion factor for private consumption 3 and the nominal exchange rate. This ratio measures price level differences between two countries (in our case between a European country and the United States) and can be expressed as follows:

US in each European country-i.e. using the relative CPIs corrected by the exchange rate variations. More specifically, we construct the monthly domestic price level series of country i on the period from January 1970 to July 2011 as follows: 4 P i,t = P P P i,2005 N ER i,2005 × CP I i,t CP I i,2005 N ER i,t N ER i,2005 × CP I U S,t CP I U S,2005 × 100 (7) where i = 1, ..., 12 denotes the European country. P P P i,2005 is the PPP for private consumption for country i relative to the US in 2005 (euros per US dollar). CP I i,t , CP I i,2005 , CP I U S,t and CP I U S,2005 are respectively the country i's CPI at time t and at year 2005, and the US CPI at time t and at year 2005. N ER i,2005 is the euro/dollar exchange rate in year 2005. 2005

Table 1 :

 1 Cointegration tests results

		1970.01-1987.06	1987.07-1998.12	1999.01-2011.07
		p-value Half-life	p-value	Half-life	p-value	Half-life
	Austria	0.0700*	48	0.0001***	0.2	0.6677
	Belgium	0.3824		0.2513		0.8373
	Finland	0.7334		0.7609		0.1358
	France	0.0661*	2	0.5410		0.3531
	Greece	0.1180		0.5343		0.9717
	Ireland	0.6870		0.5132		0.1236
	Italy	0.3395		0.6533		0.1715
	Luxembourg 0.3450		0.1222		0.7061
	Netherlands	0.3198		0.3442		0.0969*	1.6
	Portugal	0.1846		0.0402**	9.8	0.0050***	12.5
	Spain	0.3441		0.6428		0.0008***	5.5

*** (resp. **, *): rejection of the null hypothesis of no cointegration at the 1% (resp. 5%, 10%) significance level. Half-lives are expressed in years. Source:

[START_REF] Guerreiro | On price convergence in Eurozone[END_REF]

.

Table 2 :

 2 Second-generation panel unit root tests (p-values)

		Whole	Core	Northern	Southern
		sample		Periphery	Periphery
	1970.01-1987.06
	Specification	C	C	T+C	C
	CIPS	0.41	0.95	0.7	0.42
	Choi
	P m	0***	0***	0.79	0***
	Z	0***	0***	0.71	0.21
	L *	0***	0***	0.69	0***
	1987.07-1998.12
	Specification	C	C	T+C	C
	CIPS	0.25	0.01***	0.99	0.99
	Choi
	P m	0***	0***	0.73	0.01***
	Z	0***	0***	0***	0.02**
	L *	0***	0***	0.59	0.02**
	1999.01-2011.07
	Specification	C	T+C	T+C	T+C
	CIPS	0.84	0.5	0.99	0.24
	Choi
	P m	0.01***	0***	0.7	0.85
	Z	0.01***	0***	0.56	0.91
	L *	0.01***	0***	0.55	0.9

* (resp **, ***): rejection of the unit root null hypothesis at the 10% (resp. 5%, 1%) significance level. C (resp T+C) denotes model with individual effects (resp. individual effects and time trends). (a) For the CIPS test, all statistics are based on univariate AR(p) specifications with p ≤ 8. (b) For the Choi's test, the optimal lag orders in the individual ERS statistics

[START_REF] Elliott | Efficient tests for an autoregressive unit root[END_REF]

for each series are determined with p ≤ 8; under the null hypothesis the Choi's statistics are standard Normal when T and N converge jointly to infinity.

Table 3 :

 3 Pooled Mean Group estimates of the error-correction term and estimated half-lives

		Whole Core	Northern	Southern
		sample		Periphery	Periphery
	1970.01-1987.06
	φ	-0.069 -0.063	No	-0.112
	Half-life	9.69	10.65 cointegration	5.83
	1987.07-1998.12
	φ	-0.109 -0.157	No	-0.067
	Half-life	6	4.06	cointegration	9.99
	1999.01-2011.07
	φ	-0.055 -0.067	No	No
	Half-life	12.25	9.99	cointegration cointegration

φ denotes the error-correction term estimated by the PMG method. Half-lives are expressed in months.

Table 4 :

 4 Pooled Mean Group estimates of the error-correction term and estimated half-lives

	Countries		1970.01-1987.06 1987.07-1998.12 1999.01-2011.07
	Austria	φ Half-life	-0.006 115	-0.37 1.5	-0.05 13.51
	Belgium	φ Half-life	-0.08 8.31	-0.14 4.59	-0.01 68.97
	France	φ Half-life	-0.13 4.97	-0.05 13.51	-0.15 4.26
	Italy	φ Half-life	-0.02 34.31	-0.03 22.75	-0.02 34.31
	Luxembourg	φ Half-life	-0.07 9.55	-0.19 3.29	-0.08 8.31
	Netherlands	φ Half-life	-0.04 16.38	-0.13 4.98	-0.07 9.55
	Greece	φ Half-life	-0.15 4.26	-0.08 8.31	No cointegration
	Portugal	φ Half-life	-0.11 5.95	-0.05 13.51	No cointegration
	Spain	φ Half-life	-0.07 9.55	-0.04 16.98	No cointegration
	φ denotes the error-correction term estimated by the PMG method. Half-lives are expressed
	in months.

Table 5 :

 5 GDP PPP per capita relative to Germany in percent between 1970 and 2010

	Countries	1970	1987	1998	2007	2010
	Austria	94.12 99.95 102.8 107.2	105.8
	Belgium	96.77 97.22 98.38 100.4	98.27
	Finland	80.93 90.26 86.73 100.2	94.93
	France	94.96 94.73 92.38 91.53	88.74
	Greece	74.18	68.5	65.5	78.6	72.49
	Ireland	62.09 65.74 98.55 122.9 107.65
	Italy	86.08	92.9	91.14 86.11	80
	Luxembourg 145.3 150.3 184.5 221.95	205
	Netherlands 112.6 102.07 108.5 112.5	110.7
	Portugal	52.98 57.62 68.01 65.84	64.86
	Spain	74.92 74.21 80.03 85.39	80.6

Authors' calculation on the basis of OECD data.

Table 6 :

 6 Technological specialization in 1987Technological specialization in -1998Technological specialization in and 1999Technological specialization in -2009

	Specialization

Table 7 :

 7 Definition of CITI indexes

	Index	Definition
	15	food products and beverages
	16	tobacco products
	17	textiles
	18	wearing apparel ; dressing and dyeing of fur
	19	dressing of leather ; luggage
	20	woods and products of wood and cork (except furnitures)
	21	paper and paper products
	22	publishing, printing and reproduction of recorded media
	23	coke, refined petroleum products and nuclear fuel
	24	chemical products
	25	rubber and plastic products
	26	non metallic mineral products
	27	basic metals
	28	fabricated metal products (except machinery and equipments)
	29	machinery and equipment n.e.c.
	30	office, accounting and computing machinery
	31	electrical machinery and apparatus n.e.c.
	32	radio and television communication equipment
	33	medical, precision and optical instruments
	34	motor vehicles, trailers and semi-trailers
	35	other transport equipment
	36	furniture, manufacturing n.e.c.

Source : CITI, OECD.

Namely, EU, ECB, and IMF.

We will evidence in Section 2 that this methodology is very questionable since CPI do not account for differences in price levels.

Following the World Bank definition, the PPP conversion factor for private consumption is the number of units of a country's currency required to buy the same amount of goods and services in the domestic market as a US dollar would buy in the United States, the conversion factor being applicable to private consumption.

See Guerreiro and Mignon (2011). Note that to construct the price level series, it is assumed that the reference goods basket has remained the same over the 1970-1995 period.

P P P series are extracted from the OECD database. NER and CPI series are from IFS, except the German and the Irish CPIs that come from Datastream.

Note that since our price levels have a common benchmark, the US, our series already account for nominal exchange-rate variations.

In a previous version of the paper, a Eurozone price series was constructed, and PPP was investigated between each domestic price level series and the aggregate price level series of the 11 respective partners. The results were similar to those reported here, illustrating the robustness of our results to the choice of the

[START_REF] Johansen | Statistical analysis of cointegration vectors[END_REF][START_REF] Johansen | Estimation and hypothesis testing cointegration vectors in Gaussian vector autoregressive models[END_REF]'s tests have also been applied, leading to the same conclusions.

At the 1% significance level, the null hypothesis of no cointegration is rejected for Portugal and Spain at the end of the period, and for Austria during the second sub-period. Note however that half-lives for Portugal and Spain are quite long, meaning that reversion to PPP, when it exists, takes a long time.

An alternative classification would have been to retain the distinction between core and peripheral countries, the latter group being composed by Greece, Ireland, Portugal and Spain. Our choosing classification presents the advantage of distinguishing between Northern and Southern non-core countries.

See Hurlin and Mignon (2006) and[START_REF] Hurlin | What would Nelson and Plosser find had they used panel unit root tests?[END_REF] for a detailed presentation of panel unit root tests.

Economically, cross-section dependence can arise for several reasons, such as socioeconomic interactions and common factors. Cross correlation can also cause size distortions to first-generation panel tests. The cross-section dependence has been investigated using the CD test developed by[START_REF] Pesaran | General diagnostic tests for cross-section dependence in panels[END_REF] showing that such correlations exist in our samples of countries (results are available upon request to the authors).

Matlab codes (Version 7.00) provided by Christophe Hurlin are used to implement these second-generation panel unit root tests (http://www.univ-orleans.fr/deg/masters/ESA/CH/churlin_R.htm).

This hypothesis can be tested using an Hausman-type test.

Appendix

The definition of the technological specialization is based on the revealed comparative advantages for trade micro-indicators in the manufacturing category (CITI database, available on OECD). Table 7 reports the sector associated to each number.

On the basis of the analysis of these indexes at a disaggregated level, we consider the following typology:

• Specialization in low technology sectors: indexes 15 to 21.

• Specialization in medium technology sectors: indexes 22 to 28 and 36.

• Specialization in high technology sectors: indexes 29 to 35. 1987[START_REF] Frankel | The endogeneity of the optimum currency area criteria[END_REF][START_REF] Pesaran | Pooled mean group estimation of dynamic heterogeneous panels[END_REF]-2009 Austria 17, 19, 20, 21, 22, 25, 27, 15, 16, 19, 20, 21, 22, 25, 28, 29, 31, 34, 36 26, 27, 28, 29, 31, 34, 36 Belgium 15, 17, 23, 24, 25, 26, 27, 15, 17, 21, 23, 24, 25, 26, 34 27, 34 Finland 20, 21, 22, 27, 28, 29, 31, 32, 20, 21, 22, 27, 28, 29, 31, 35, 36 32, 35, 36 France 15, 22, 23, 24, 25, 26, 28, 15, 21, 22, 23, 24, 25, 26, 31, 34, 35 28, 29, 31, 33, 34, 35 Germany 21, 24, 25, 26, 29, 31, 33, 16, 21, 24, 25, 29, 31, 33, 34, 35 34, 35 Greece 15, 16, 17, 18, 26, 27 15, 16, 17, 18, 22, 25, 26, 27, 28 Ireland 15, 22, 24, 30 15, 22, 24, 30 Italy 17, 18, 19, 26, 28, 29 15, 17, 18, 19, 21, 26, 28, 29 Netherlands 16, 22, 23, 24, 25, 30, 33 16, 22, 23, 24, 30, 33 Portugal 15, 17, 18, 19, 20, 21, 26, 15, 16, 17, 18, 19, 20, 21, 28, 31, 34, 36 23, 26, 28, 31, 34, 36 Spain 15, 19, 23, 25, 26, 27, 28, 15, 19, 21, 23, 24, 25, 26, 27, 34 28, 34


Robert D Cairns
email: robert.cairns@mcgill.ca

Vincent Martinet
email: vin-cent.martinet@grignon.inra.fr

An Environmental-Economic Measure of Sustainable Development

Keywords: sustained development, growth, maximin, sustainability indicator

A central issue in the study of sustainable development is the interplay of growth and sacrifice in a dynamic economy. This paper investigates the relationship among current consumption, growth, and sustained consumption in two canonical, stylized economies and in a more general context. It is found that the maximin value measures what is sustainable and provides the limit to growth. Maximin value is interpreted as an environmental-economic carrying capacity and current consumption or utility as an environmental-economic footprint. The time derivative of maximin value is interpreted as net investment in sustainability improvement. It is called durable savings to distinguish it from genuine savings, usually computed with discounted utilitarian prices.

Introduction

Sustained development is a phrase that describes growth out of poverty toward a developed state that can be sustained for what Solow (1993) calls the very long run.

In an efficient economy, growth or development entails the diversion of resources from consumption by the current generation to investment that will increase productivity in the future. For development to be sustainable, the path followed by the economy must be within environmental and technological constraints.

Although the implications for a poor society are not usually stressed, a policy proposal of greater current investment and less consumption has been advanced in several economic models that assume efficiency in the attainment of a specified goal, usually maximizing discounted-utilitarian welfare. The current standard of living in a less developed country may be so low, however, that one cannot contemplate reducing it. Sacrificing the interests of the present may be inconsistent with the Brundtland Report's (World Commission on Environment and Development, 1987) famous dictum on sustainability, which balances and protects the interest of the present as well as the future. [START_REF] Asheim | Net National Product as an Indicator of Sustainability[END_REF] Optimal growth theory, however, neither specifies the extent of sacrifice envisaged, nor values growth per se in the definition of welfare, although growth is considered good.

The question of how to express the notion of sustained development formally has been partially addressed in other contexts in economics. Growth theorists have specified as parameters certain variables that could have been modeled as choices.

Among them are a constant savings ratio, a constant capital-output ratio, balanced growth, or a constant "bliss" level of utility. Holding a variable constant in this way has simplified complicated dynamic problems and has allowed for many revealing [START_REF] Asheim | Net National Product as an Indicator of Sustainability[END_REF] One way to avoid this invidious trade-off is to assume that, while the present generation is poor, there is some possibility of improvement from the base of the present. Llavador et al. (2010), for example, find that sustainable consumption for the USA was higher than actual consumption in 2000. A possible reason is inefficiency. As Llavador et al. indicate, the long-term solution is to address the inefficiency, not necessarily to invest more in the present. In the present paper, we are not focusing on inefficiency, but on investment.

analyses.

Other analysts have found tentative evidence that there is a preference among consumers for wage or consumption profiles that increase through time (Lowenstein and Sicherman, 1991; Frank and Hutchens, 1993). Examples used are of consumption growing at a constant rate over an individual's lifetime. If such preferences can be applied to a whole society, growth of consumption at a constant rate can be considered to be a generalization of a sustained path in that its "distribution over time has some definite standard shape" (Hicks, 1946: 184).

The present paper addresses the implications of a conscious choice by a society between current sacrifice and growth. The issue is how to grow out of poverty, to improve what can be sustained. Following the tentatives in growth theory and in positive economics, we assume that the economies are not pursuing a specific objective but rather a parametric policy that seems plausible, for example, constant growth or constant employment. Sustainable development means that an acceptable standard of living is reached in the long run and then sustained. We examine the conditions for the given growth pattern to be sustainable.

Our study is motivated by two model economies that have been prominent in the study of sustainability. In a simple fishery, a fish stock is harvested and consumed directly. Open access leads to overexploitation. At any point, however, the society is assumed to be in a position to choose a level of employment in the industry, and hence of forbearance in exploiting the stock. At the beginning of the program the stock is at a low level (is "overfished") and the society wishes to rebuild its stock by limiting current consumption. In the steady state, the harvest is equal to natural growth and is thus sustained.

In the Dasgupta-Heal-Solow (DHS) model of an economy dependent on manufactured capital and an essential, non-renewable resource (Dasgupta and Heal, 1974; Solow, 1974), sustaining consumption at a constant level requires that investment in manufactured capital offset the depletion of the resource (Hartwick, 1977). A deviation downward from that possible constant-consumption path can allow for growth at a parametric rate through investment (d'Autume and Schubert, 2008; Asheim et al., 2007). The economy can choose from many different paths of sustained development. Overshooting the sustainable path is also possible in the model.

Since much of the discussion of sustainability in economics has been done in terms of simple models, especially these two models, it is natural for the paper to pass from the particular to the general. The findings from the simple models are the basis of a generalization to more complicated economies.

The Setting

Each of the two canonical models addresses a fundamental issue in environmental economics. Each implies that growth is subject to environmental constraints. Open access in the fishery leads to a tragedy of the commons. The DHS economy illustrates the fact that sustaining an economy may not involve a steady state. Each of open access and growth can lead to unsustainability and to a poverty trap.

Our analysis is based on a modification to the maximin program that allows for growth. A maximin path maximizes the standard of living of the poorest generation, looking forward from the present (Cairns and Long 2006). What is sustained (supported from below) along a feasible path of the economy is the minimum level of consumption of any generation over the very long run. The maximum attainable such minimum level, or the maximin level, is what is sustainable. Let social utility at time t be represented by U (t). The sustainable or maximin level of utility at time t in a dynamic economy is given by max Ū s.t. U (s) ≥ Ū ∀s ≥ t.

(

If the economy pursues the maximin objective in a regular maximin problem, the standard of living remains constant over the indefinite future (Burmeister and Hammond, 1977; Cairns and Long, 2006). [START_REF] Asheim | Can NNP be used for welfare comparisons?[END_REF] This means that if a planner decides to apply 2 A comparison with "strong" sustainability is in order. Strong sustainability is the capacity to ensure a minimum standard of, or a minimum level of an index of, environmental quality.

Maintaining a higher level of the index is considered more desirable. A maximin program could the criterion immediately in a poor economy, future generations may be mired in a "poverty trap" involving continuing levels of the standard of living equal to the low level of the present: poverty may be sustained. The criticism implies that the present generation is considered to be at a level of poverty that is so dire that the future must be rescued from it. Our modification to the maximin program is a response to this criticism. Since, for a regular maximin program, the constant utility path is a Pareto-efficient solution, we show that for growth to occur the standard of living of the present must be reduced to an even lower level than that of the poverty trap. The path envisaged is one in which the society chooses a growth pattern beginning from its current, low level toward a higher, sustained level. We describe the trade-offs among present consumption, growth and long-run sustained consumption.

For the sake of definiteness, utility or the standard of living of the society is frequently interpreted herein as its level of consumption, broadly defined. We argue that the maximin level of consumption is a representation of sustainability, as it

gives the highest consumption level that can be sustained from the current economic state. Even though a maximin policy may not be being pursued, at any economic state a maximin level of consumption can be determined by solving the maximin problem for the stocks at that state. The evolution of this maximin value along any On the one hand, genuine savings extends the concept of savings in the national accounts to include changes in the quantities of capital goods, especially environmental goods, that do not have market prices. [START_REF] Asheim | Constant Savings Rates and Quasi-Arithmetic Population Growth under Exhaustible Resource Constraints[END_REF] It is equal to the current change in social welfare, which is usually defined to be the integral of discounted social utility. Non-negative genuine savings is sometimes considered to be an indicator of sustainability because current welfare does not decrease. If genuine savings are non-negative it is, however, not possible to say whether welfare will be sustained in the long-run (Asheim, 1994). Even if negative genuine savings means that the current utility is not sustainable, the opposite is not true (Pezzey, 2004). The welfare integral can increase at the current moment but eventually decrease, even if the environment is incorporated into optimal decisions (Dasgupta and Heal, 1979).

Moreover, on the trajectory being followed by the economy, the change in social welfare may be negative over a short period of time, but then turn upward. Genuine savings with a discounted utility objective function is not the long-run measure sought in considering sustainability.

On the other hand, the ecological footprint has been proposed as an indicator of the environmental limit to sustainable output. It seeks to compare the level of current utilization of environmental resources (i.e., the ecological footprint) with the available flow of environmental services (i.e., the ecological carrying capacity), evaluated in terms of land of a given quality. If the level of utilization is greater than the flow of available services, the society depletes the stock and is considered to be unsustainable at its current level of utilization. is as much a concern as immediate growth.

The Economics of Unsustainability

In this section, we illustrate how it is possible to use the maximin value to characterize the unsustainability of a development path in the two canonical economies.

The Simple Fishery

We

Ṡ (t) = S(t) [1 -S(t)] -S(t)E(t). (2)
We assume that the effort

E belongs to the interval [0, 1]. The open-access regime has E(t) = E 0 = 1.
In this model, the highest sustainable level of consumption is called the "maximum sustainable yield" (MSY). Its value is

C M SY = max S [S (1 -S)] = 1 4 .
The associated stock is S M SY = 1 2 and the equilibrium level of effort is E M SY = 1 2 . In this model, the MSY stock is a benchmark for both ecological and economic overexploitation. [START_REF] Burmeister | Maximin Paths of Heterogeneous Capital Accumulation and the Instability of Paradoxical Steady States[END_REF] If the initial state S 0 is lower than that associated with the MSY, the maximin criterion (1) leads to a constant harvest in equilibrium, C(t) = S 0 (1 -S 0 ). If the initial state is above the MSY level, the maximin value is the MSY harvest. We thus define the maximin value, given the state S at time t, of this economic system as

m(S) = S M SY (1 -S M SY ) if S > S M SY , S(1 -S) if S ≤ S M SY . (3)
If S ≤ S M SY , the level of effort, E mm , on a maximin path is such that the harvest is equal to the natural growth, so that E mm S = S (1 -S), or E mm = 1 -S.

We shall now consider a consumption path that exhausts the resource and is thus unsustainable. For the sake of simplicity, we suppose that S (0) = 1 and that the resource is initially in open access, i.e., the effort level is E 0 = 1. That level of effort is maintained so long as there is a net benefit to fishing. [START_REF] Cairns | Maximin: A Direct Approach to Sustainability[END_REF] The dynamics of the exploited resource becomes

Ṡ (t) = S(t) [1 -S(t)] -S(t) = -[S (t)] 2
The stock evolves as

S(t) = 1 1 + t . ( 4
)
Consumption, C(t) = E 0 S (t) = 1/ (1 + t), decreases toward zero as the stock decreases toward zero.

To characterize the sustainability of this unsustained path we study the evolution of the maximin value. During a first period (t ∈ [0, 1]), until the stock falls to the level S (1) = S M SY = 1 2 , the stock decreases but is still above S M SY . The maximin value is thus constant at the MSY level. After t = 1, the maximin value decreases as the stock decreases below S M SY = [START_REF] Asheim | Net National Product as an Indicator of Sustainability[END_REF] 2 . An analytical expression of the maximin value as a function of time is easily derived from equation (4) for any t ≥ 1:

m (S(t)) = S (t) [1 -S (t)] = t (1 + t) 2 .
( that the resource stock is initially at its carrying capacity, and the initial catch level C 0 is greater than the Maximum Sustainable Yield. As the resource stock is large, it is initially easy to catch C 0 , but as the stock is depleted, more effort is needed. Defining the fishing effort as a feedback rule depending on C 0 and S(t), one generates a constant harvesting trajectory, with decreasing stock and increasing effort. After some time, the effort cannot increase above its upper limit. The system then reaches the open-access equilibrium, with decreasing catches.

figure illustrates an important result: once the MSY stock is overshot, the maximin value decreases. We interpret this decrease as an indicator of overshooting of the environmental capacity to provide fish over the very long run.

The Dasgupta-Heal-Solow Model

Consider a society that has stocks of a non-renewable resource, S 0 , and of a manufactured capital good, K 0 , at its disposal at time t = 0. It produces output (consumption c and investment K) by use of the capital stock and by depleting the resource stock at rate

r (t) = -Ṡ (t) , (6)
according to a Cobb-Douglas production function:

c + K = F (K, r) = K α r β , with 0 < β < α < α + β ≤ 1. ( 7
)
This model has been used by many authors to study the implications of exhaustibility of an essential resource, including how to sustain consumption in the face of exhaustibility. If the discounted-utility criterion is applied to this economy, consumption decreases asymptotically toward zero (Dasgupta and Heal 1974, 1979).

Analysis of how consumption can be sustained requires a different approach from discounted utilitarianism. For given levels of the capital and resource stocks, Solow (1974) and Dasgupta and Heal (1979) show that the maximal consumption that the economy can sustain, the maximin level, is given by

m(S, K) = (1 -β) (α -β) β 1-β S β 1-β K α-β 1-β . ( 8
)
Since this aggregate of the two stocks measures the capacity of the economy to sustain the standard of living m(S, K) for the long term, we interpret it as the index of sustainability. It is an increasing function of both stocks. Let the initial level of consumption be

c (0) = c 0 < m (S 0 , K 0 ) .
To illustrate the economics of unsustainability in this model, we assume that the society unmindfully pursues growth at a constant rate g > 0, so that the consumption path is

c(t) = c 0 e gt . (9)
To complete the set of decision rules constituting the resource-allocation mechanism of the society, we choose a rate of resource use determined as follows:

r(K, S) = (α -β) 1 1-β S 1 1-β K -1-α 1-β . ( 10
)
This rule is the optimal feedback rule in a maximin program and is discussed more formally in the next section.

We study the evolution of the maximin value m[S(t), K(t)] along the path defined by eqs. ( 9) and (10). Fig. 2 (Exponential consumption and Maximin value function) presents a growth path for a given level of c 0 and a value of g.

The development path can be described by three phases.

1. The first phase is before the point labeled "overshooting time." Since at the outset the level of consumption is less than the maximin value m(S 0 , K 0 ), the capital stock can be built faster than along the hypothetical maximin path starting from the same initial state, on which c(t) = m(S 0 , K 0 ), and the sustainable consumption of the economy, i.e., the maximin indicator m (S(t), K(t)), can increase over time.

2. Once the consumption c(t) is greater than the maximin value m(S(t), K(t)), the maximin value decreases, meaning that the sustainable productive capacity of the economy decreases. Consumption growth in this second phase can still, however, be myopically pursued until a consumption peak when the consumption level reaches the total production level and investment is zero.

3. The third phase starts after this consumption peak. Consumption is bounded above by the decreasing level of production. There is no more investment. Consumption decreases toward zero, and remains above the currently sustainable level given by eq. ( 8). This example illustrates an overshooting of long-term productive or environmental-economic carrying capacity represented by the maximin value. Con-sumption represents the economic footprint. As long as the footprint is lower than the carrying capacity, the latter can increase over time as a result of investment in productive capacity. Once the footprint is higher than the productive capacity, however, the decrease of the maximin value indicates an unsustainable development path.

The analysis of these two models suggests that the maximin value can be used as an indicator of unsustainability even when the policy is not to sustain the economy by following the maximin path. Unsustainability occurs when the maximin value decreases. In the next section, we examine how it can characterize sustained development paths in the same two illustrative models. Later we show in fairly general models that it has the same properties along any development path that satisfies certain conditions.

Sustained Development

In this section, we examine the conditions for a development path to be sustainable.

In the two models under study, we assume that a given growth pattern is pursued. Sustainable development is defined as follows. Consumption increases according to the assumed growth pattern as long as it is lower than the maximin value, which represents generalized economic carrying capacity. When consumption catches up the maximin value, the economy stops growing and follows the maximin path starting from the economic state reached. As the maximin value is dynamic, there is a trade-off between initial consumption, the pursued growth rate, and the duration of the growth period or equivalently the level of sustained consumption that is reached in the long run.

The Simple Fishery

Let the initial state S 0 be lower than the MSY biomass, i.e., S 0 < S M SY , as may have occurred if the economy has been facing a "tragedy of the commons" for some time because of an initial open access to the resource. The stock can be considered to be over-exploited, or vulnerable to over-exploitation, and a poverty trap. If the stock recovers from over-exploitation, the maximin value can increase.

Let a level of effort be chosen and remain constant at the level E 0 ∈]0, 1[. Such a strategy could aim at increasing the available resource and sustainable consumption while maintaining an acceptable level of employment in the fishery. Consumption is given by C(t) = E 0 S(t) and the dynamics of the exploited resource becomes

Ṡ (t) = S(t) (1 -E 0 -S(t)) . ( 11
)
Along this trajectory, the stock evolves as

S(t) = 1 1 1-E 0 + 1 S 0 -1 1-E 0 e -(1-E 0 )t . ( 12
)
The system tends toward a limit, S ∞ = 1 -E 0 .

The rule of constant effort completely determines the trajectory of this fishery. By equation ( 3), when S ≤ S M SY , the maximin level of effort is given by

E mm (S) = 1 -S.
This level of effort maintains the stock at a stationary level that may correspond to a "poverty trap." In order to recover from a period of overfishing, society must harvest less than the maximin harvest m (S) = S (1 -S) so that the stock can grow and the maximin value function can increase along the trajectory. This feature of the problem illustrates that there is no "free lunch" for the future. Current effort must be less than E mm , and consumption less than

C mm = S 0 (1 -S 0 ).
Under a strategy of constant fishing effort, with E(t) = E 0 < 1 -S 0 = E mm (S 0 ), fish consumption increases with the stock size. Fig. 3 depicts the following trajectories through time:

• The natural growth of the stock from the initial state S 0 = 0.1 (without harvesting). 7

• The growth of the resource stock with constant fishing effort E 0 = E M SY = 1 2 . The stock tends toward S M SY . This trajectory is labeled "stock recovery." [START_REF] Cairns | Sustained Development of a Society with a Renewable Resource[END_REF] With no consumption (C(t) = 0, i.e., E(t) = 0), the dynamics of the resource stock is given

• The trajectory of the maximin value function along the trajectory for E 0 = 1 2 . The maximin value increases toward the MSY level.

• The consumption pattern, which increases as the stock increases and catches up to the maximin value. Consumption tends toward the MSY. We stress that the recovery of the fishery (and thus the increase in consumption) is possible only because consumption is lower than the maximin level at all times.

The long-run consumption depends on the reduction of the present consumption, the constant fishing effort being between the maximin value E mm (S 0 ) = 1 -S 0 and the MSY value E M SY = 1 2 . A lower fishing effort, and hence current consumption, by

S(t) = 1 1 + e -t ( 1 S0 -1) . ( 13
)
The stock recovers faster, but present generation does not consume at all. entails a higher long-run consumption. [START_REF] Chichilnisky | The Green Golden Rule[END_REF] Fig. 4 presents the trajectories of maximin value and catches for three different recovery strategies (for three different effort levels) with, again, an initial fish stock S 0 = 0.1. For this stock, the initial maximin value is 0.1(1 -0.1) = 0.09. • The first strategy (trajectories denoted by M 0.9 and C 0.9 ) corresponds to a constant fishing effort E 0 = E mm = 0.9. At this effort level, the stock is in equilibrium at the initial value, i.e., S ∞ = S 0 = 0.1. The harvest is equal to the maximin value from the initial stock at all times.

• The second strategy (trajectories denoted by M 0.7 and C 0.7 ) corresponds to a constant fishing effort E 0 = 0.7 < 0.9. The fish stock increases asymptotically toward a limit, S ∞ = 1-E 0 = 0.3 (not represented on the figure). The harvest increases toward the maximin harvest for this stock, S ∞ (1-S ∞ ) = 0.21, which is lower than the MSY.

• The third strategy (trajectories denoted by M 0.5 and C 0.5 ) is that depicted in Fig. 3, with the fishing effort set constant at the MSY equilibrium effort, 0.5.

The maximin value increases asymptotically toward the MSY value and the harvest increases toward the MSY, which is 0.25.

There is a non-linear relationship between C 0 and C ∞ which is determined by the chosen (constant) effort level. Recovery effort belongs to [E M SY , E mm (S 0 )]. If the effort is small and equal to

E M SY , present consumption is low (C 0 = E M SY S 0 )
and the limiting consumption is the MSY. If the effort is equal to E mm (S 0 ), the stock remains at the initial level S 0 , and the present and limiting consumption are equal. (There is no growth.) This is the maximin path, sometimes criticized as possibly entrenching a poverty trap. Intermediate cases are defined according to the relationship

C ∞ = lim t→∞ E 0 S (t) = E 0 (1 -E 0 ) = C 0 S 0 1 - C 0 S 0 , ( 14
) for C 0 ∈ [S 0 /2, S 0 ], i.e., for E 0 ∈ [1/2, 1]
. The possibility frontier between present and future consumption is described by Fig. 5.

Any pair (C 0 , C ∞ ) that is achievable with constant effort belongs to this frontier. Social preferences between present and future consumption can be given by a function Ψ (C 0 , C ∞ ), which can be maximized along the frontier. Several particular solutions are represented in Fig. 5

The Dasgupta-Heal-Solow Model

Sustained development in the DHS model can be represented as follows. Suppose that the economy chooses an initial level of consumption c 0 that is less than the sustainability indicator provided by the maximin value m (S 0 , K 0 ). This choice allows for growth. We assume that the social planer aims at pursuing consumption growth at a constant rate g > 0 until some time T when it reaches the maximin level m(S(T ), K(T )). [START_REF] Dasgupta | The Welfare Economic Theory of Green National Accounts[END_REF] Let us introduce formally the way our problem deviates from the maximin problem, and the resulting resource-allocation mechanism. In a maximin problem, the objective is mathematically expressed as the maximization of the Hamiltonian H(c, r, S, K) = λ Ṡ + µ K subject to the constraint c (t) ≥ c, where c is the maximin consumption (Cairns and Long, 2006). It is equivalent to maximize the Lagrangean:

T E E E 0 C ∞ C 0 = E 0 S
L(c, r, S, K, ν) = H(c, r, S, K) + ν (c -c) = λ Ṡ + µ K + ν (c -c) .
Note that the term ν (c -c) corresponds to the complementarity slackness condition, and is always equal to zero. Cairns and Long (2006, Proposition 1) show that the co-state variables of a maximin problem, λ and µ, are equal to the derivatives of the maximin value function with respect to the state variables, i.e., λ = ∂m ∂K and µ = ∂m ∂S . One thus has ṁ (S, K) = λ Ṡ + µ K. The previous expression of the Lagrangean is thus equivalent to

L(c, r, S, K, ν) = ṁ (S, K) + ν (c -c) .
The problem is tantamount to maximizing the net investment at maximin shadow values ṁ (S, K) subject to the constraint that consumption is no less than the maximin value. In the maximin problem, this maximin value is a parameter of the optimization, and it is increased as much as possible. Hartwick's (1977) rule is that, at the maximum, H(c, r, S, K) = ṁ (S, K) = λ Ṡ + µ K = 0. The minimal consumption c is increased up to the point at which the maximal net investment is nil.

We here deviate from this maximin optimization problem in the sense that we do not maximize the minimal consumption over time. On the contrary, we consider a given consumption pattern. We assume, however, that the social planer does not waste sustainability improvement, and maximizes net investment accounted at the maximin shadow values subject to the consumption pattern constraint. [START_REF] Dasgupta | The Optimal Depletion of Exhaustible Resources[END_REF] In the present problem, the constraint is that c (t) = c (t) = c 0 e gt and the aim is to maximize the Lagrangean,

L(c, r, S, K, ν) = ṁ(S, K) + ν (c -c) , = - ∂m ∂S r + ∂m ∂K (K α r β -c) + ν c -c 0 e gt ,
That is to say, the program is to maximize ṁ (S, K) subject to the modified constraint. [START_REF] Dasgupta | The Economics of Exhaustible Resources[END_REF] Resource-allocation mechanism. The society chooses the level of extraction of the resource that maximizes ṁ, the net investment at the maximin shadow values, conditional on the consumption path.

As the consumption pattern is given by equation

c (t) = c 0 e gt (15)
this resource allocation mechanism defines the natural resource extraction. By differentiating the maximin value function (eq. 8) logarithmically with respect to time t, we express the rate of growth of the maximin value as

ṁ m = α -β 1 -β K K + β 1 -β Ṡ S = α -β 1 -β K K - β 1 -β r S . ( 16
)
Using this derivative we compute the extraction rule r(K, S) that maximizes the rate of growth of the maximin value (whatever is the consumption), and find that it is given by r(K, S) = (α -β)

1 1-β S 1 1-β K -1-α 1-β . ( 17
)
This feedback rule is the same as the one along the maximin path, for which growth of both consumption and maximin value are zero. For this resource-allocation mechanism, the consumption side of the economy is determined by the growth pattern, and the production side is determine so as to maximize the instantaneous gain of sustainable utility.

The limits to growth. There is an endogenous limit to the time for which growth can be supported at rate g, given the described resource allocation mechanism. In fact, the long-run level of consumption is endogenous. To avoid the unsustainable type of trajectory described in previous section, the economy must switch at some given, the two others can be derived.

For any initial pair (S 0 , K 0 ) 0, there is a maximin level of consumption m 0 = m (S 0 , K 0 ) > 0 given by equation (8). Also, for any initial pair of stocks it is possible at time t = 0 to choose any pair

(c 0 , g) ∈ A (S 0 , K 0 ) {]0, m 0 [ × ]0, ∞[ ∪ (m 0 , 0)}
The path in which (c 0 , g) = (m 0 , 0) is the maximin (sustained) path. It has no growth. A path in which (c 0 , g) ∈ {]0, m 0 [ × ]0, ∞[ } (so that g > 0 and c 0 < m 0 ) has growth. However, growth at a constant rate cannot go on forever. [START_REF] Dasgupta | The Environment and Emerging Development Issues[END_REF] Let us define the endogenous time T (S 0 , K 0 , c 0 , g) at which consumption catches up to the dynamic maximin value indicator. We have

c(T (•)) = c 0 e gT (•) = m [S (T (•)) , K (T (•))] .
From then on, growth is no longer sustainable, and the level of consumption must remain at the maximin level; i.e., for t ≥ T (.), sustainability implies that c (t) = m (S (T ) , K (T )).

At time T , only the part of the resource-allocation mechanism that drives the level of consumption changes, from allowing consumption to grow at rate g to keeping consumption constant at c ∞ = m (S (T ) , K (T )). Resource use is still determined by the maximin efficiency definition.

One may view society as making a choice according to a preference ordering P (c 0 , g, c ∞ ), by which initial consumption, the rate of growth and the very longrun, sustained consumption are evaluated. Fig. 7 depicts a convex-concave correspondence from the initial pair (S 0 , K 0 ) to the attainable frontier, B (S 0 , K 0 ) [START_REF] Dasgupta | Net National Product, Wealth and Social Well Being[END_REF] For a given growth rate, a lower level of initial consumption allows a higher long-run level. For a given initial consumption, a lower growth rate allows a higher long-run consumption (as the actual consumption catches the maximin level more slowly).

{(c 0 , g, c ∞ ) feasible from (S 0 , K 0 )}. Growth is possible only if c 0 < m(S 0 , K 0 ).
Given the initial level of consumption c 0 , there is a trade-off between the eventual maximin consumption that is sustained after time T and the rate of growth that is sustained up to that level. A level of present consumption that is closer to the maximin value m(S 0 , K 0 ) entails a lower prospect for growth.

A General Measure of Sustainability

The general trade-off envisaged in the present paper is between social utility at time t and the ultimate utility reached in the long run. The long-run level is endogenous when one has chosen a growth pattern, as is illustrated in both models above. In the tradition of Ramsey's (1928) model of undiscounted utility, some authors assume [START_REF] Dasgupta | Net National Product, Wealth and Social Well Being[END_REF] Negative growth (g < 0) is required if c 0 > m(S 0 , K 0 ). (This is not represented on the figure.) . The exogenous bliss level of utility coincides with the (green) golden rule and is approached asymptotically. Our approach contrasts with this view. In our fishery model the bliss level is the MSY, which is not necessarily the long-run level chosen by the society. In the DHS model there is no exogenous bliss level and the long-run consumption is also a social choice.

In the examples above, changes in the indicator, the maximin value, have a clear meaning in terms of the sustainability of the society along the chosen trajectory.

If consumption is lower than the maximin value over some interval and the output so freed up is invested in long-term productive capacity, the sustainable level of consumption can be permanently increased. Conversely, consumption over a current interval can be increased at the expense of investment and hence of sustainable consumption in the future.

The present section examines the generalization of this conclusion to more general problems. Consider a vector of capital stocks X ∈ R n + . The transition equation for each element X i , i = 1, . . . , n is given by Ẋi = F i (X, c) .

where c ∈ C (X) ⊆ R p represent a vector of decision within the set C (X) of admissible controls at state X. The maximin value is denoted by m (X) ∈ R + . For any feasible set of controls c = (c 1 , ..., c n ), the evolution of the maximin value is given by

µ (X, c) = ṁ (X) | c = n i=1 Ẋi ∂m (X) ∂X i = n i=1 F i (X, c) ∂m (X) ∂X i .
The terms ∂m(X) ∂X i are the maximin shadow values of state X, and depend only on the current state and not on the economic decisions. They are thus defined whatever the economic trajectory given by functions F i (X, c).

A sustainable growth program is defined as follows.

Definition 1 Maximization of sustainability improvement. The resource-allocation mechanism is said to maximize sustainability improvement at each instant if decisions c maximize the increase of the maximin value subject to the given growth pattern:

c maximizes µ (X, c) = n i=1 F i (X, c) ∂m (X) ∂X i (18)
s.t. U (X, c) = Ū (t)
The interpretation of this resource allocation mechanism is that, at each instant, the current generation increases the limit to growth as much as possible, given its current utility defined by the assumed growth pattern.

We aim to prove that, if sustainability improvement is maximized, having a lower level of utility than the maximin utility leads to an increase in the maximin value. [START_REF] Doyen | Maximin, Viability and Sustainability, EconomiX Working Papers No 2010-19[END_REF] Note that the resulting path is not necessarily efficient in the usual sense. [START_REF] Frank | Wages, Seniority, and the Demand for Rising Consumption Profiles[END_REF] This does not, however, diminish the importance of the present result. We prove that, for the growth to be sustainable, there must be a sacrifice of present utility with respect to the maximin value, and that this necessary condition is also sufficient if the resources freed up by the reduction of utility are invested optimally to increase the productive capacity of the economy.

We provide a proof for problems satisfying the following assumptions.

Assumption 1. There is a j ∈ {1, . . . , p} such that, given a state vector X and a decision vector c = (c 1 , . . . , c j , . . . , c p ), one has U c j > 0.

Assumption 1 means that there is at least one control variable which influences utility continuously around a given level. We assume that the control has a positive effect on utility for the sake of simplicity, i.e., U c j > 0. Our result, however, holds for a negative effect, by redefining c j as -c j . Note that when Assumption 1 is satisfied, there is an interval [c j , cj ], such that U (c 1 , . . . , c j , . . . , c p ), X < U ((c 1 , . . . , cj , . . . , c p ), X).

Assumption 2. For all capital stocks X i , i = 1, ..., n,

∂m (X) ∂X i ∂F i (X, c) ∂c j ≤ 0 . (19)
Assumption 2 means that the control c j does not increase (decrease) investment in stocks having a positive (negative) contribution to the maximin value.

Assumption 3. For at least one capital stock (think of "manufactured capital" for concreteness), the condition in Assumption 2 is strictly satisfied:

∂m (X) ∂X i ∂F i (X, c) ∂c j < 0 . (20)
Assumption 3 means that the control c j has an effect on the maximin value.

A simple example is if the control c j is consumption in the DHS model, consumption increases utility and comes from forgone investment in manufactured capital. Investment in manufactured capital contributes directly to the maximin value and does not affect the change in the resource.

These assumptions are, in fact, quite general. If assumption 1 is not satisfied, utility does not depend on the decisions at the considered economic state.

Assume that Assumption 2 is not satisfied, i.e., there is some i ∈ {1, . . . , n} such that ∂m(X) ∂X i ∂F i (X,c) ∂c j > 0, while Assumption 1 is. It would then be possible to increase both current utility and the maximin value by increasing c j . In that case, the maximin problem has no solution as there would be always one control increasing utility and the maximin. Assumptions 1 and 2 entail that there is no free lunch in a maximin problem.

Assume that Assumption 3 is not satisfied while Assumptions 1 and 2 are, i.e, there is a value c

i j ∈ (c j , c m j ) if cj < c m j , or c i j ∈ (c m j , cj ) if cj > c m j such that F i (X,c)-F i (X,c m ) (cj-c m j ) = ∂F i( X,c i ) ∂c j . Eq. (22) becomes µ (X, c) -µ (X, c m (X)) = n i=1 ∂F i (X, c i ) ∂c j cj -c m j ∂m (X) ∂X i = cj -c m j n i=1 ∂F i (X, c i ) ∂c j ∂m (X) ∂X i .
By Assumptions 2 and 3, the sum term is strictly negative. Since, by Assumption 1, U c j > 0, one necessarily has that cj < c m j since Ū (t) < m (X(t)), and thus cj -c m j < 0. The product of these two negative terms is thus positive, and one concludes that µ (X, c) -µ (X, c m (X)) > 0. Since µ (X(t), c m (X(t))) ≥ 0, one deduces that µ (X(t), c) > 0. 17

Under the resource-allocation mechanism that maximizes sustainability improvement, the path satisfies the following problem:

max c∈C(X(t)) n i=1 F i (X(t), c) ∂m (X(t)) ∂X i . (23)
Since c ∈ C (X(t)), one has max c∈C(X(t))

µ (X (t) , c) ≥ µ (X(t), c) > 0.

QED.

The In the regular part of the fishery, the control, E, can be used to increase or Since the maximin value m is a function of the stocks X, its change through time on any path,

µ (X, c) = i ∂m (X) ∂X i Ẋi ,
is a weighted measure of change in the stocks. Sustainable growth entails positive net investment when evaluated at the sustainability prices ∂m/∂X. The current level of well being is not being sustained if µ (X, c) < 0, i.e., if net investment at sustainability prices is negative. The proposition confirms that a policy of sustainable growth in an efficient program costs the current generation as compared to pursuing a maximin policy; sustainable growth occurs only if u < m. There is no free lunch for the future.

The criterion involving net investment closely resembles the instantaneous criterion that has erroneously been applied to genuine savings or genuine investment as determined from a green extension of the national accounts (e.g., World Bank, 2006; Dasgupta, 2009). Green accounting is an improvement to the traditional national accounts in that it generalizes them to included non-marketed goods. The issue regarding sustainability, however, turns not solely on the assets to be included but also on the shadow or accounting prices at which investment is evaluated. We disagree with the World Bank (2006: 41) when they write, "Economic theory tells us that there is a strong link between changes in wealth and the sustainability of development-if a country (or a household, for that matter) is running down its assets, it is not on a sustainable path. For the link to hold, however, the notion of wealth must be truly comprehensive." It is not enough for the notion of wealth to be comprehensive (to include all assets, not just marketed assets). In sustainability analysis it is equally vital to get the accounting prices right. An increase of the integral of discounted utility implies that genuine savings, computed at competitive prices, are positive at a given instant. However, constancy or increase of welfare signaled by nonnegative genuine savings may not be lasting or durable.

Rather, the genuine savings indicator can be positive along a competitive path even though consumption exceeds the maximin level (Asheim, 1994). Welfare measured as discounted utility may ultimately turn downward in spite of the positive, current, genuine savings (Pezzey, 2004). Even though it is comprehensive, genuine savings as it is usually computed, i.e., using discounted utilitarian prices, is not equal to investment evaluated at the shadow values of the maximin program, i ∂m(X) ∂X i

Ẋi .

We distinguish genuine investment, be it applied to maximized social welfare or the level of welfare generated by the resource-allocation mechanism of a real economy, from investment calculated from the maximin value by calling the latter sustaining or durable investment.

Durable investment is the indicator of the current change in sustainability.

It is comprehensive investment evaluated at maximin shadow prices, along any particular path of the economy. It is the statistic that is appropriate in expressing sustainability improvement. For sustained development at t the economy must have both u (t) ≥ 0 and ṁ (t) ≥ 0. This last condition means that the maximal sustainable utility, i.e., the set of sustainable utility opportunities for future generations, increases at the current time. Current growth does not jeopardize the capacity of future generations to sustain utility.

According to the generalized concept of genuine savings indicator formalized by Asheim (2007), non-negative net investment (accounted at the shadow values of a given welfare function) is associated with non-decreasing welfare at the current time.

There is, however, no normative reason to have a non-negative net investment when welfare is defined as discounted utility. Discounted utility does not require nonnegative investment. Maximin does. Non-negative investment at maximin prices is a characteristic of maximin paths, and thus of the maximin value function. Pursuing non-negative investment at maximin prices, even in a sub-optimal economy, is consistent with sustainability and with the optimality concept of maximin. Pursuing non-negative investment at discounted utility prices is not a criterion for sustainability and is inconsistent with the optimality concept of discounted utility.

Conclusion

The discussion stresses a property of a growth path that is not stressed by proponents of sustainable growth out of poverty. If the maximin path is not pursued, but instead some growth path is followed, then earlier generations must be deprived in order to divert toward investment the resources needed to sustain growth. Whether this deprivation is consistent with the vague criterion enunciated in the Brundtland report in terms of "needs" is not obvious. Growth is possible only at a cost. Open access, which in abstract terms is the main environmental problem facing humanity, is an inefficiency that cannot be overcome without current sacrifice. Growth is possible only within limits given by the technology and the environment. Otherwise, it can cause overshooting.

We come to affirm the conclusion drawn by Solow (1993: 172): From an empirical point of view it makes sense to approach sustainability from the dual, that is to say, to use the approach of the footprint rather than the one based on the national A dynamic and fully comprehensive footprint, using physical measures, would be dual to a measure based on prices. Pricing in units of land can be interpreted as a pricing system that is equivalent to pricing with a specific numeraire. Maximin analysis puts the insights of the ecological footprint on a sounder, more comprehensive footing, based not on land capacity but on "generalized capacity to produce economic well-being" (Solow, 1993).

The definition of durable savings "works" for any resource-allocation mechanism.

But durable savings must be evaluated at "the right prices", the durable (or sustaining) prices. If there is a suspicion that the market is not producing a sustainable result, then market prices are likely wrong. How to get the prices is a difficult question, even in very simple models. The difficulty is no reason to use genuine savings with discounted utilitarian prices to measure long-term sustainability. This practice can be misleading and send an incorrect message as genuine savings can be positive even if current utility exceeds the maximal sustainable utility (Asheim, 1994; Pezzey, 2004), and the maximin value indicator is decreasing.

The indicator of sustainability on any program, optimal or not, is the maximin value. Durable investment, the change in the maximin value, is the indicator of whether or not the level of well-being that can be sustained is increasing or decreasing.

Fig. 1 (

 1 Fig. 1 (Open Access trajectory of the fishery) presents the evolution of fish stock, consumption rate, and maximin value over time under open access. The 6 A different story, which would have resulted in the same outcome, is the following. Consider

SMSYFigure 1 :

 1 Figure 1: Open Access trajectory of the fishery

Figure 2 :

 2 Figure 2: Exponential consumption and Maximin value function

Figure 3 :

 3 Figure 3: Evolution of the maximin value function along a constant effort trajectory leading to Maximum Sustainable Yield

Figure 4 :

 4 Figure 4: Sensitivity analysis (with respect to the constant effort level)

  , including the Green Golden Rule (Chichilnisky, Heal and Beltratti, 1995) corresponding to Ψ (C 0 , C ∞ ) ≡ C ∞ ; myopic behavior from open access, corresponding to Ψ (C 0 , C ∞ ) ≡ C 0 ; and the maximin, corresponding to Ψ (C 0 , C ∞ ) ≡ [min(C 0 , C ∞ )]. Once initial and final consumption are chosen, the (logistic) growth rate is endogenous under the assumption that effort is constant.

Figure 5 :

 5 Figure 5: Trade-off between present consumption and long-run consumption in a fishery with constant effort and S 0 = 0.1.

  time T from the constant-growth path to a maximin path characterized by constant consumption c ∞ ≡ m(S(T ), K(T )). The program is an open-loop path, determined at time 0. Fig. 6 illustrates such a sustained-development path.

Figure 6 :

 6 Figure 6: Exponential consumption and Maximin value function

Figure 7 :

 7 Figure 7: Necessary trade-offs between initial consumption, growth rate, and longrun (sustained) consumption in the DHS model

Proposition 1

 1 i = 1, ..., n. It would then be possible to increase the utility at current time without modifying the maximin value. The problem is non-regular in this state (Cairns and Tian 2010, Martinet and Doyen 2011). Taken together, Assumptions 1-3 simply mean that our result is valid on the regular parts of maximin problems, i.e., around states for which the maximin value is affected by the current decisions. On an trajectory with maximal sustainability improvement in which Assumptions 1-3 hold, sgn ṁ (X(t)) = sgn [m (X (t)) -Ū (t)].

  to decrease the level of consumption, C (S, E) = SE. Moreover, ∂C (S, E) /∂E = S > 0, and ∂F (S, E) /∂E = ∂ [S (1 -S) -SE] /∂E = -S < 0. The proposition applies.In the DHS model there are two stocks, resource S and manufactured capital K, and two controls, consumption c and extraction r. Utility is given by U (S, K, c, r) = c. Moreover, one has (∂F 1 /∂c) (∂U/∂c) < 0. Reducing consumption allows for an increase in the capital stock and an increase in the maximin value. Again, the proposition applies.

  accounts. The reason has to do with the prices obtained from extending national accounting toward green accounting. The prices of green accounting, which are the shadow values for discounted utility, are not the "right" accounting prices. The right accounting prices are the maximin shadow values, which are based on what is sustainable. The ecological footprint uses physical measures that can be measured correctly. Through its set of explicit trade-offs that make land the numeraire, ecological footprint analysis has implied a form of substitutability among natural and other stocks. The ecological footprint has no explicit objective, although an implicit objective is to sustain the society. This lack of an explicit objective is what leads to the derivation of accounting prices from the (natural) constraints facing the society.

  path of the economy is said to be a sustainable development at time t if

	17 If Ū (t) was not feasible by modifying only one control, one must iterate the reasoning to show
	the successive additional improvements of maximin value with respect to the previous vector of
	decisions, until Ū (t) is reached, and the associated control belongs to C (X(t)). See footnote 16.

ṁ | t ≥ 0 and u (t) ≥ 0. Under Assumptions 1 and 2, and given the maximization of sustainability improvement, a necessary and sufficient condition for sustainable development is that u (t) ≤ m | t . This condition, applied to the maximin value, is what Pezzey (1997) uses to define sustainability. Sustainable utility decreases ( ṁ | t < 0) if u (t) > m | t .

The comprehensive vector of capital stocks accounted for is then the same as the vector of capital stocks used to define the maximin value. The value of each stock is, however, different.

This model is often written using the parameters r, S sup and q to represent the natural growth rate of the resource, its carrying capacity, and the catchability of the resource, so thatṠ(t) = rS(t) 1 -S(t) S sup -qS(t)E(t).In our model, without loss of generality we define units of time, of effort and the resource such that r = 1, S sup = 1, and q = 1. The expressions are less cumbersome, but one must be careful to keep track of the units in which the variables are measured.

[START_REF] Burmeister | Maximin Paths of Heterogeneous Capital Accumulation and the Instability of Paradoxical Steady States[END_REF] We do not consider the cost of effort, for sake of simplicity. It implies that the Maximum Economic Yield (golden rule) coincides with the Maximum Sustainable Yield.

are not considered as they would results in lower catches both for present and future generations.

Another possibility is to imagine a path for which the rate of growth smoothly approaches the maximin value. For example, the path followed could be a logistic growth curve. This path would be more difficult to solve than the path proposed in the text but would give no more insight into the problem.

We interpret this objective and the associated resource allocation mechanism in the general model of next section.

Note that the shadow values are the same as that of the maximin problem, and that the modified complementarity slackness condition is also equal to zero.

From a theoretical point of view, the described framework could be used to give a rigorous meaning to "sustainable degrowth" from an initial consumption larger than the maximin value and a negative growth rate.

The opposite result, i.e., having a higher level of utility than the maximin utility leads to an decrease in the maximin value, is straightforward from the usual maximin problem.

In particular, it does not maximize the long-run utility given the initial consumption and growth pattern. The definition of such an efficient sustainable growth path is a task for future research.

Proof of Proposition 1 Denote by c m (X (t)) = c m 1 (X(t)) , . . . , c m j (X(t)) , ..., c m p (X(t)) the controls associated with the hypothetical maximin program starting from the stocks X(t) at time t. Since the maximin value does not decrease in a maximin program, for these controls the change of the maximin value in that problem, µ (X(t), c m (X(t))), is non-negative and given by

Let the state at time t be given and the chosen utility level be equal to Ū (t) < m (X(t)), exogenously fixed by a chosen development pattern. Consider the set of decisions C (X(t)) that make it possible to attain exactly utility Ū (t):

Under Assumption 1, there is a vector of decisions c = (c m 1 , . . . , cj , . . . , c m n ) ∈ C (X(t)) that achieves the utility constraints and differs from the maximin decisions c m (X(t)) only for the j th control. [START_REF] Hartwick | Intergenerational Equity and the Investing of Rents from Exhaustible Resources[END_REF] If the decisions c are applied, then

Comparing this expression with µ (X(t), c m (X(t))) yields

As the functions F i (X, c) are continuous and differentiable in c j , we know from the mean value theorem (Lagrange's finite-increment theorem) that, for i = 1, . . . , n, [START_REF] Hartwick | Intergenerational Equity and the Investing of Rents from Exhaustible Resources[END_REF] If it is not possible to modify c j sufficiently to reduce utility to level Ū (t), i.e., if cj / ∈ [c j , cj ], one needs to proceed by steps, and take the intermediate value U (c m 1 , . . . , c j , . . . , c m p ), X as a reference (instead of Ū (t)) in the previous definition of admissible states and in what follows, and repeat the process (with a new control) until utility is reduced to Ū (t).