Balázs Égert
email: balazs.egert@oecd.org.

Jorgen Elmeskov

Robert Ford

David Heald

Phil Hemmings

Edouardo Olaberria

Artur Radziwill

Urban Sila

Douglas Sutherland

Jaejoon Woo

Volker Ziemann

Public debt, economic growth and nonlinear effects: Myth or reality?

Keywords: E6, F3, F4, N4 public debt, economic growth, nonlinearity, threshold effects 1. OECD Economics

 were the first to point out that a public debt-to-GDP ratio higher than 90% of GDP is associated with considerably lower economic performance in advanced and emerging economies alike. A string of recent empirical papers broadly validates this threshold value. This paper seeks to contribute to this literature by putting a variant of the Reinhart-Rogoff dataset to a formal econometric testing. Using nonlinear threshold models, there is some evidence in favour of a negative nonlinear relationship between debt and growth. But these results are very sensitive to the time dimension and country coverage considered, data frequency (annual data vs. multiyear averages) and assumptions on the minimum number of observations required in each nonlinear regime. In addition, we also show that nonlinear effects can kick in at much lower levels of public debt (between 20% and 60% of GDP). These results, based on bivariate regressions on secular time series, are largely confirmed on a shorter dataset (1960-2010) when using a multivariate growth framework that accounts for traditional drivers of long-term economic growth and model uncertainty. Nonlinear effects might be more complex and difficult to model than previously thought. Instability might be a result of nonlinear effects changing over time, across countries and economic conditions. Further research is certainly needed to fully understand the link between public debt and growth.

Introduction

The financial and economic crisis prompted by the unwinding US subprime mortgage market resulted in deep economic recession in many countries of the world. Governments and central banks reacted to the Great Recession by firing heavy artillery: fiscal and monetary policy expansion, unprecedented in size and in the way they were coordinated across countries, were swiftly enacted in advanced and emerging markets, and huge banking sector bailouts prevented the collapse of the financial system. While these actions certainly helped smooth the cycle, discretionary fiscal loosening and banking sector bail-outs contributed to a large extent to a sharp increase in many countries' public debt-to-GDP ratio. It is against this background that Reinhart and Rogoff (2010) pointed out the existence of strong negative effects of high public debt on economic growth. Using simple descriptive statistics, they demonstrated forcefully that economic growth slows down considerably if the public debt-to-GDP ratio exceeds 90%.

There are a number of channels through which public debt is likely to hamper long-term growth. First, tax hikes needed to service a higher public debt crowd out private investment by reducing disposable income and saving, raise the distortionary costs of taxation, and are likely to result in non-neutral tax treatment within and across asset classes, thus amplifying distortions. Second, soaring public debt will push up long-term sovereign yields in a nonlinear fashion, as the likelihood of default increases. High long-term rates crowd out productive public investment, and, more importantly, reduce private investment by increasing the cost of capital. Reduced investment in R&D will have long-lasting negative impacts on growth [START_REF] Elmeskov | Post-Crisis Debt Overhang: Growth and Implications across Countries[END_REF]. Third, public authorities, especially in countries with weak institutions, may decide to inflate away debt, and high inflation has a notoriously detrimental effect on growth [START_REF] Kumar | Public debt and growth[END_REF].

Many recent empirical papers sought to pin down and explain the possibly nonlinear negative relationship between public debt and growth. Most of these papers broadly confirm that the turning point beyond which economic growth slows down sharply is around 90% of GDP. [START_REF] Cecchetti | The real effects of debt[END_REF] find a threshold of 86% of GDP for a panel of 18 OECD countries and for the period from 1980 to 2010. [START_REF] Padoan | Avoiding debt traps: financial backstops and structural reforms[END_REF] report similar effects for a similar group of countries but a longer period (1960 to 2010). Covering a mix of advanced and emerging market economies, [START_REF] Kumar | Public debt and growth[END_REF] finds a turning point at 90% of GDP. [START_REF] Checherita | The impact of high and growing government debt on economic growth: An empirical investigation for the euro area[END_REF] and [START_REF] Baum | Debt and growth: new evidence for the euro area[END_REF] report similar results for a set of euro area countries. [START_REF] Caner | Finding the tipping pointwhen sovereign debt turns bad[END_REF] and [START_REF] Elmeskov | Post-Crisis Debt Overhang: Growth and Implications across Countries[END_REF] show that the tipping point is probably lower: 77% for a set of 77 countries, and 66% for a dozen of OECD countries, respectively. Finally, in a recent contribution, [START_REF] Panizza | Public debt and economic growth: is there a causal effect?[END_REF] argue that a negative correlation between debt and growth does not imply causality, as lower growth can result in a higher public debt to GDP ratio.

This paper seeks to contribute to this literature by putting a variant of the Reinhart-Rogoff dataset to a formal econometric testing by first using the thresholds proposed by [START_REF] Reinhart | Growth in a time of debt[END_REF] and then identifying the thresholds endogenously on the basis of the testing procedure proposed by [START_REF] Hansen | Threshold Effects in Non-Dynamic Panels: Estimation, Testing and Inference[END_REF] for the period 1790 to 2009 and 1946 to 2009. We then embed the growth-debt relation in a general multivariate growth framework and combine it with Bayesian model averaging to gauge the impact of model uncertainty on the presence of threshold effects for 1960 to 2010.

We find some evidence in favour of a negative nonlinear relationship between debt and growth. But these results are very sensitive to the time dimension and country coverage considered, data frequency (annual data vs. multi-year averages) and assumptions on the minimum number of observations required in each nonlinear regime. We also show that nonlinear effects are likely to kick in at much lower levels of public debt (between 20% and 60% of GDP). These results, based on bivariate regressions on secular time series, are largely confirmed on a shorter dataset when using a multivariate growth framework that accounts for traditional drivers of longterm economic growth and model uncertainty.

The paper is organised as follows. Section 2 provides some stylised facts about the public debtgrowth nexus by focusing on the Reinhart-Rogoff dataset. Section 3 presents empirical results for the variant of the Reinhart-Rogoff dataset. Section 4 embeds the debt-growth relationship in a multivariate growth framework and provides results using Bayesian model averaging. Finally, section 5 summarises and provides some policy implications.

Stylised facts

In their influential paper, [START_REF] Reinhart | Growth in a time of debt[END_REF] rely on descriptive statistics to show that public debt as a share of GDP may have a detrimental effect on the rate of growth of real GDP. More specifically, they argue that the crucial level of public debt is 90%, beyond which growth slows down considerably. Their charts and tables are meant to prove this claim: average and median annual GDP growth rates are shown relative to the level of the central government debtto-GDP ratio for the period from 1946 to 2009. For a group of selected advanced countries, average GDP growth drops from around 3% to below 2% as public debt passes the threshold of 90% of GDP. The fall is more dramatic if growth is measured in terms of the median, rather than the average: a public debt-to-GDP ratio higher than 90% is associated with zero GDP growth. A similar pattern can be observed if only data for the US are considered: public debt exceeding the threshold of 90% goes in tandem with a decline in annual growth from about 3.5% to well below zero. The drop is again more pronounced if the median and not the average growth rate is looked at. The conclusion is strikingly similar for a group of selected emerging market economies, for which growth slows down by an annual 2 percentage points with public debt moving from below to above 90% of GDP. Figure 1 below reproduces these results for the group of advanced and emerging market economies. While [START_REF] Reinhart | Growth in a time of debt[END_REF] do not provide the data used in their paper, the data on central government debt can be obtained from the data appendix of another paper of the same authors [START_REF] Reinhart | From financial crash to debt crisis[END_REF]. Real GDP growth rates are available for a number of countries for the same time period from the Barro-Ursúa macroeconomic dataset (Barro and Ursúa, 2011). Putting these two datasets together enables us to replicate broadly the the Reinhart and Rogoff data coverage on selected advanced economies. But there are some differences. First, our data excludes Ireland and includes Switzerland. Second, the emerging market country coverage of our data differs substantially from theirs . We have data on GDP growth for 16 out of the 24 countries included in their empirical investigation. But we also have data for five additional developing countries. Finally, and importantly, our series often start considerably earlier. This gives us more observations for the full sample period, whereas our coverage is almost identical for the period 1946-2009. Table A1 in the appendix lists the differences.

We use our dataset to replicate and extend the results of [START_REF] Reinhart | Growth in a time of debt[END_REF]. Figures hereafter show average and median real GDP growth as the central government debt-to-GDP ratio varies for our entire sample, the group of advanced economies and two groups of emerging markets, the first including the 16 countries covered in [START_REF] Reinhart | Growth in a time of debt[END_REF] and the second containing all emerging countries for which data are available. In addition to the whole sample and the post-war period , we also show numbers for the period from 1790 to 1939.

Several conclusions can be drawn from Figure 2, which shows GDP growth along rises in public debt as a share of GDP. First, for all countries and for the group of developed countries, growth remains broadly stable as public debt increase from the range of 60% to 90% of GDP to above 90%. This suggests the absence of any sudden change (fall) in growth rates beyond 90% of the public debt-to-GDP ratio. As a matter of fact, for these two samples, growth rates appear to decline gradually with the rise in public debt from the range 0% to 30% to above 90%. Growth even seems to increase slightly once the debt ratio is above 90% for some periods. Second, for the period 1946-2009, growth slows down for the two groups of emerging market economies as public debt moves from the range of 60% to 90% to beyond 90% of GDP. But GDP growth rates associated with debt levels above 90% are higher than when debt varies between 30% and 60% of GDP. Finally, contrary to [START_REF] Reinhart | Growth in a time of debt[END_REF], some of these observations are not very sensitive to the alternative use of the median of annual growth rates and the average growth rate. The difference only matters for emerging markets and for the whole and the pre-war periods (1790-2009 and 1790-1939): for the group of emerging markets, higher debt goes in tandem with a increase in the median growth rate (Panel C and D of figure 2).

We carry out a sensitivity analysis to see the extent to which the average growth rates are subject to the influence of individual countries. Jackknifing the sample of the selected advanced economies (that is recalculating the average growth rates by omitting one country at a time) indicates that the average for the whole advanced economy sample is not influenced by outlier countries if the debt ratio is below 90%. But the uncertainty is relatively large when debt is higher than 90% of GDP: the gap between the minimum and maximum of the average growth rate is one percentage point for the entire sample period (1790-2009) and half a percentage point for the two sub-periods (Figure 3).

Another way of having a preliminary idea on the correlation between growth and debt is to plot the annual GDP growth rates against the debt ratio. The scatter plots presented in Figure 4 are striking: GDP growth and the public debt ratio do not appear to have any relationship with one another for any of the country groups and sub-periods considered. The general picture does not change if public debt is plotted with a lag of one year (Figure A1 in the appendix).

Multi-year averages eliminate cyclical and other short-term effects, which may contaminate the scatter plots of annual figures. Therefore, Figure 4b plots non-overlapping 5-, 8-and 10-year averages for growth and public debt for the period 1946-2009. Yet eyeball econometrics does not suggest an apparent negative correlation between debt and growth, especially for the full sample and the group of emerging market economies. Using debt with a one period lag confirms these observations (Figure A2 in the appendix).

A formal test of nonlinearity

Econometric issues

We apply a multi-step approach to our secular dataset covering the period from 1790 to 2009 to formally analyse the link between public debt and growth. We first start by looking at the bivariate linear relationship by estimating the following regression for growth and debt:

(1)

where is annual real GDP growth and debt stands for the central government debt-to-GDP ratio. Equation ( 1) is estimated for a pooled panel and with country fixed effects.

We then estimate threshold models, in which the effect of debt on growth depends on the level of debt. In these models, the thresholds put forth by [START_REF] Reinhart | Growth in a time of debt[END_REF] are imposed. A tworegime model is estimated where the coefficient on public debt can be different below and above the threshold of 90% of the debt ratio. We also estimate a three-regime model, in which the three regimes are separated by the 60% and 90% debt thresholds. Finally, a four-regime model account for all 3 thresholds proposed by Reinhart and Rogoff: 30%, 60% and 90% of central government debt. This four-regime model can be written as follows:

_10_D_GDP _10_CDEBT                        % 90 % 90 % 60 % 60 % 30 % 30 4 4 3 3 2 2 1 1 DEBT if DEBT DEBT if DEBT DEBT if DEBT DEBT if DEBT y t t t t t t t t t             (2)
A shortcoming of this approach is that the choice of the number of the regimes and the value of the thresholds are necessarily arbitrary and we do not know whether any of the nonlinear models provides a better fit for the underlying data than alternative linear and nonlinear specifications. The testing procedure developed by [START_REF] Hansen | Threshold Effects in Non-Dynamic Panels: Estimation, Testing and Inference[END_REF] helps solve these problems because it first determines the threshold values endogenously through a grid search, and second, it tests the different models sequentially against one another using bootstrapping methods. The linear specification is tested against a two-regime model. If the null hypothesis of the linear model can be rejected against the alternative of a two-regime model, the null of a two-regime model is tested against the alternative of a three-regime model. The two-regime and three-regime models can be written as follows.

             T DEBT if DEBT T DEBT if DEBT y t t t t t       2 2 1 1 (3a)                  2 3 3 1 2 2 2 1 1 1 T DEBT if DEBT T DEBT T if DEBT T DEBT if DEBT y t t t t t t t          (3b)
T is the value of the threshold of debt in the two-regime model and T 1 and T 2 are the lower and upper threshold values of debt in the three-regime model. A grid search with steps of 1% of the distribution is carried out to find the value of the threshold variable (public debt) that minimizes the sum of squared residuals of the estimated two-regime model. The grid search starts at 20% of the distribution and stops at 80% to ensure that a sufficient number of observations falls into each regime.

The three-regime model is estimated based on two threshold values of the threshold variable that minimise the sum of squared residuals across the estimated models. The threshold from the tworegime model is held fixed and a grid search is used to identify the second threshold. We impose the restriction that the two thresholds should be separated at least by 20% of our sample observations. Once the second threshold is identified, a backward grid search is performed to identify the first threshold as suggested by [START_REF] Hansen | Threshold Effects in Non-Dynamic Panels: Estimation, Testing and Inference[END_REF].

We can proceed with the sequential testing of the models, once the thresholds are identified. [START_REF] Hansen | Threshold Effects in Non-Dynamic Panels: Estimation, Testing and Inference[END_REF] shows that the null hypothesis of 2 1    from equations (3a) can be tested using a likelihood ratio test. Given that the likelihood ratio test statistic does not follow a standard asymptotic distribution as the threshold value is not identified under the null hypothesis, the distribution of the test statistic is obtained through bootstrapping with random draws with replacement. The bootstrap test was carried out using N=500 replications. If the likelihood ratio test statistic rejects the null hypothesis of the linear model against the two-regime model (on the basis of the bootstrapped critical values), whether there are three different regimes rather than only two regimes is also analysed. The bootstrap procedure described above is applied to the tworegime and three-regime models.

Estimation results

Simple bivariate panel regressions yield a negative link between growth and public debt. The coefficient is always negative but its size is not particularly large in economic terms: a 10 percentage increase in the public debt ratio is associated with 0.1 to 0.2 percent lower economic growth. In addition, the statistical significance of this result varies very much across different country samples and time periods. Results tend to be statistically significant for the whole period and for the post-war period but not for 1790 to 1939. The result is also sensitive to country coverage: for the two sub-periods (1790-1939 and 1946-2010), the estimate is not significant for the smaller sample of emerging markets but it is when five countries are added to the sample (Table 1).

As for the nonlinear specifications estimated using threshold values taken from [START_REF] Reinhart | Growth in a time of debt[END_REF], the results again show some instability. The estimated coefficients are not significant at the 10% level for 1790-1939 for the samples including all countries and the advanced countries only and for 1946-2009 for the smaller group of emerging countries.

While the estimated coefficients of the public debt ratio variable is almost always negative for 1790-2009 and 1946-2009, indicating that higher debt relates to slower growth, the size of the coefficients decreases by a factor of 2 to 5 as the debt ratio rises. This could imply that the harmful effect of public debt on growth diminishes with rising debt, but it could also well be the case that lower coefficients indicate that a one percentage point increase in the public debt-to-GDP ratio means a lower rate of growth of debt for higher levels of debt. We re-run the equations using the rate of growth rather than the level of the debt-to-GDP ratio as independent (nonlinear) variable2 . For the two-and three regime models, it indeed seems to be the case that the negative coefficients of the growth rate of debt increases with a rise of the level of debt. But for the 4regime models, the negative coefficient for debt ranging from 60% to 90% of GDP is lower than if debt is lower than 60% or higher than 90% (Table 1).

A serious problem with the correlation between public debt and growth is that any change in the growth rate of real GDP will have a mechanical effect on the debt-to-GDP ratio. Therefore, we reestimate the nonlinear specifications using the lagged public debt-to-GDP ratio. Indeed, the previous results can be confirmed only for two sub-samples: for all countries from 1946 to 2009 and for the advanced country group from 1946 to 2009. The statistically significant negative nonlinear relationship between debt and growth disappears for the other sub-samples (Table A1 in the appendix). Again, we check whether a decrease in the negative coefficients on the level of the debt-to-GDP ratio implies a declining or increasing negative impact as the level of debt rises. Astonishingly, the results show the absence of any negative correlation between the growth rate of debt and economic growth for debt levels exceeding 90% of GDP. The only exception is the group of advanced countries for 1946 to 2009.

These disappointing results may be due to fact that the choice of the debt thresholds is arbitrary. So it is natural to try to figure out the values of the thresholds in a data-driven approach. There is evidence for nonlinear effects both for the entire period (1790-2009) and for the post-war period for the relation between growth and contemporaneous debt. The estimated negative coefficients tend to decrease for higher debt levels. But an important difference is that the debt thresholds are very low: slightly below 20% of GDP for the sample encompassing all countries and slightly above 30% for the group of emerging markets. The thresholds picked in the threeregime model for advanced economies are also considerably lower than the 90% put forward by [START_REF] Reinhart | Growth in a time of debt[END_REF]: the first threshold is about 30% and the second threshold is around 60% GDP (Table A2 in the appendix).

But again, the evidence obtained for the debt ratio lagged with one year is much less convincing. Table 2 shows that while the estimated coefficients are negative for the high-debt regime for the entire sample and the developed country group (1790-2009), they are not statistically significant and their size is very small. For the same period, no nonlinearity could be detected for emerging market economies. When the time coverage is restricted to 1946 to 2009, the tests of nonlinearity show that the relation between debt and growth differ depending on the level of debt. For the advanced countries, a low level of debt is associated with a nontrivial positive effect of debt on growth and with a small negative impact above the debt threshold of 20% of GDP: a 10 percentage point rise in the debt-to-GDP ratio goes in tandem with 0.08 percent decline in economic growth. For the other country groups, the coefficient estimates are statistically insignificant in the high-debt regime (Table 2). Results obtained using the growth rate of the public debt-to-GDP ratio are not much more reassuring. It is only for the period 1790 to 2009 and for developed countries that a high level of debt is associated with a lower growth performance (Table 3): a 1 percent increase in the debt ratio goes hand in hand with an almost 0.4 percent drop in economic growth. Yet this result cannot be confirmed for the period 1946-2009 for which the null hypothesis of a linear specification cannot be rejected against the alternative of threshold nonlinearities (Table 3).

Using non-overlapping multi-year averages can potentially eliminate short-term noises from our data series. Results based on such data for advanced countries are reported in Table 4. There is no empirical evidence for a negative nonlinear relation between debt and growth if we use 5-year averages. For 8-year averages, regressions based on the growth rate of debt indicate strong nonlinear effects (Table 4): no correlation between growth and debt if debt is roughly below 40% of GDP and a negative relationship above this threshold, which imply that a 1 percent increase in debt is associated with a 0.1 percent decline in growth. For 10-year averages, the negative effect is even stronger above the threshold of 67% of GDP: a 0.3 percent decline in growth. These findings are somewhat sensitive to how the threshold models are parametrised (the minimum number of observations required in each regime) but the overall conclusions remain unaltered (Table A4 in the appendix).

Finally, we also investigate whether alternative forms of nonlinearity do a better job of describing the debt-growth relation. A polynomial trend of the debt variable (equation 4) would allow for instance a smooth transition around the turning point:

(4) Ideally, a hump-shaped pattern of growth around the turning point such as plotted in Panel A of Figure 5 would be in line with a disruptive debt-growth relationship. Nevertheless, the estimation results, both for the contemporaneous and lagged level of debt, indicate that higher debt is accompanied by higher economic growth (Panel B of Figure 5). Not quite the results we were looking for. 55. The results are almost identical if the estimated parameters of the model based on lagged public debt is used..

Accounting for other drivers of growth and model uncertainty

Nonlinearity embedded in a model averaging framework

We now embed the public debt-growth nexus into a general growth framework. In accordance with a general production function approach, the level of per capita GDP (rather than real GDP growth) is a function of human and physical capital and labour input. Formulating this long-run relationship in terms of an error correction model yields a model where the rate of growth of per capita GDP depends on lagged per capita level (

) and lagged values of the other explanatory variables. Following the growth literature, physical capital can be proxied by the investment-to-GDP ratio (inv), human capital by average years of schooling (edu) and labour input by population growth (

). Additional controls used in the literature are inflation (infl) and openness (open) as additional control variables.

Hansen's threshold modelling framework can be applied to this set of covariants with a view to analysing the possible nonlinear relation between economic growth and debt:

                               T debt if debt X T debt if debt X cap t t n j t j j t t n j t j j t         1 2 1 1 1 , 2 1 1 1 1 1 , 1 (4a)                                              2 1 3 1 1 1 , 3 2 1 1 2 1 1 1 , 2 1 1 1 1 1 1 , 1 T debt if debt X T debt T if debt X T debt if debt X cap t t n j t j j t t n j t j j t t n j t j j t             (4b)
where debt is general government debt and is a vector of independent variables. But a high number of independent variables poses the problem of model uncertainty. To address this issue, Bayesian averaging of classical estimates (BACE), which provides estimates for all possible combinations of the (K) candidate explanatory variables, is given by K 2 3 . The BACE technique shows whether the inclusion of a candidate variable improves the fit of the model [START_REF] Sala-I-Martin | Determinants of Long-Run Growth: A Bayesian Averaging of Classical Estimates (BACE) approach[END_REF]. BACE determines the posterior probability attributed to each single model j M that includes the variable of interest and is conditioned on the underlying dataset (

) ( y M P j ).        K i j i T i k i T j k j j SSE T M P SSE T M P y M P 2 1 2 / 2 / 2 / 2 / ) ( ) ( ) ( ( 5
)
where SSE is the sum of squared residuals, T is the number of observations, k denotes the number of explanatory variables included in the specific model and K is the number of all explanatory variables considered. Expression (5) gives the contribution of a given model to explaining the dependent variable as compared to the other models. Expression ( 5) is then summed over the models that contain the variable of interest to obtain the posterior inclusion probability of this variable. If the posterior inclusion probability is higher than the prior inclusion probability, one can conclude that the candidate variable should be included in the estimated models. 4

The posterior mean and the square root of the variance (standard error) conditional on inclusion can be used to obtain t-statistics and to determine the significance of the individual variables upon inclusion. The posterior mean conditional on inclusion ( Or some subset of models. If the number of models to be estimated is too large, techniques such as Markov-Chain Monte-Carlo, stochastic search variable selection, or random sampling are alternative approaches to estimating all possible models. Given the relatively low number of potential explanatory variables used here, we estimate all possible combinations 4. [START_REF] Sala-I-Martin | Determinants of Long-Run Growth: A Bayesian Averaging of Classical Estimates (BACE) approach[END_REF] compare the posterior inclusion probability to a prior inclusion probability for their 67 explanatory variables in 7 variable models. The prior inclusion probability is then 7/67=0.1044.

) ( y E  ) is
regressions (even those without the variable of interest), the unconditional posterior mean of any given variable can be derived as the product of the conditional posterior mean and the posterior inclusion probability. The posterior variance of  ( ) ( y Var  ) can be calculated as follows:

       K K j j j j j j y E y M P M y Var y M P y Var 2 1 2 2 1 ) ) ( )( ( ) , ( ) ( ) (     (7)
The linear BACE approach can be extended to possible nonlinearities between growth and public debt by including equations ( 4a) and (4b) into the model space [START_REF] Crespo-Cuaresma | Nonlinearities in Cross-Country Growth Regressions: A Bayesian Averaging of Thresholds (BAT) Approach[END_REF]. In the spirit of model averaging, we estimate all possible combinations of the candidate explanatory variables. For each combination, the linear, two-regime and three-regime models are estimated. The selection between linear and nonlinear models is done by using [START_REF] Hansen | Threshold Effects in Non-Dynamic Panels: Estimation, Testing and Inference[END_REF] bootstrapping method described earlier. An advantage of this methodology is that only a single linear or nonlinear model is selected for a given set of explanatory variables.

Estimation results

We assemble a dataset, drawn from the OECD's Economic Outlook database, including nonoverlapping multi-year (5-year, 8-year and 10-year) averages for 29 OECD countries covering the period 1960-2010. As the series start for a number of countries only in the 1970s and 1980s, we also define a narrow sample including only 13 OECD countries for which time series start in the 1960s. Figure 6 below gives a flavour on the possible correlation between lagged general government debt and growth. The scatter plots do not show a clear-cut negative relationship. Bayesian averaging of classical estimates augmented for nonlinearities helps us evaluate the uncertainty stemming from model specification. Estimation results suggest the presence of a strong negative nonlinear relation between lagged general government debt and growth. For the sample of 29 OECD countries, the posterior inclusion probability is higher than 50% for the threeregime model for the 5-year, 8-year and 10-year averages (Table 4a). This indicates that the threeregime model prevails over the linear and two-regime models. But there is one exception: the debt variable, either in a linear specification or in a nonlinear fashion, is not included in the final model space for 8-year averages if each regime has to contain at least 10% of the observations. Results for the posterior mean conditional on inclusion exhibit a large amount of instability. The nature of nonlinearity differs depending on whether 5-, 8-or 10-year averages are used. For 5-year averages, there is a negative relationship between growth and debt in all three regimes, but the largest negative effect occurs when public debt is the lowest. For 8-year averages, public debt and growth have a positive correlation if the debt-to-GDP ratio is roughly below 35% but the relation turns negative above this threshold. Finally, the correlation between debt and growth is very volatile for 10-year averages: the correlation can be negative or positive depending on on how much observations are allowed in particular regimes (Table 4a).

Turning now to the group of 13 OECD countries, Table 4b shows that whether the two-or the three-regime model is selected depends on the type of the multi-annual averages (5-, 8-and 10year averages). The posterior mean is negative for 8-year averages and increases with higher public debt. For 10-year averages, the negative relationship is the largest for the low-debt regime, which contradicts the Reinhart-Rogoff prediction. Finally, for 5-year averages, there is a positive relation between debt and growth if the debt-to-GDP ratio increases. Another strong contradiction with the Reinhart-Rogoff result is that if there is a negative nonlinear link between debt and growth, they kick in at much lower debt levels: between 20% and 60% of GDP. The results are much shakier if we replace the level of the debt-to-GDP ratio by its growth rate. There is much less evidence for nonlinearity in this case, and a strong negative correlation emerges at higher levels of debt only for the group of 13 OECD countries and if 8-year averages are used (Tables 4c and4d).

Conclusions

The aim of this paper was to contribute to the empirical literature on the debt threshold beyond which negative effects kick in for economic growth. We put a variant of the Reinhart-Rogoff dataset to a formal econometric test. Using nonlinear threshold models, we found some evidence in favour of a negative nonlinear relationship between debt and growth. But these results are very sensitive to the time dimension and country coverage considered, data frequency (annual data vs. multi-year averages) and assumptions on the minimum number of observations required in each nonlinear regime. We also showed that nonlinear effects can kick in at much lower levels of public debt (between 20% and 60% of GDP). These results, based on bivariate regressions on secular time series, are largely confirmed on a shorter dataset (1960-2010) when using a multivariate growth framework that accounts for traditional drivers of long-term economic growth and model uncertainty.

Previous empirical papers, validating the Reinhart-Rogoff result of a 90% public debt ceiling beyond which economic growth slows significantly, called for debt reduction to improve longterm growth. The implications of our results are that the 90% is not a magic number. The threshold can be lower and the nonlinearity can change across different samples and specifications. Nonlinear effects might be more complex and difficult to model than previously thought. Instability might be a result of nonlinear effects changing over time, across countries and economic conditions. Further research is certainly needed to fully understand the link between public debt and growth.

Figure 1 .

 1 Figure 1. Annual real GDP growth and central government debt as a % of GDP, 1946-2009 Results of Reinhart and Rogoff (2010) developed countries emerging market economies

Figure 2 .

 2 Figure 2. Annual real GDP growth and central government debt as a % of GDP 1790-2009 1790-1939 1946-2009 Panel A. Developed and emerging market economies (41 countries)

Figure 5 .

 5 Figure 5. Polynomial functional form of nonlinearity Panel A. calibration Panel B. estimated parameters

Figure 6 .

 6 Figure 6. Non-overlapping multi-year real GDP per capita growth (DCAPPNEWOK) and lagged general government debt (GGFLQ) as a % of GDP, 1960-2010, 29 OECD countries 5-year averages 8-year averages 10-year averages

  Figure A2. Non-overlapping multi-year real GDP growth and lagged central government debt as a % of GDP, 1946-2009 Panel A. Developed and emerging market economies, 5-year averages 8-year averages 10-year averages

Table 1 . The nonlinear relation between public debt and growth, 1790-2009, annual data EXOGENOUSLY IMPOSED DEBT THRESHOLDS (30%, 60%, 90% of government debt)

 1

	ALL COUNTRIES	DEVELOPED COUNTRIES	16 EMERGING COUNTRIES	21 EMERGING COUNTRIES
	1790-
	2009

Table 2 . The nonlinear relation between public debt and growth, annual data Endogenous debt thresholds

 2

	Nonlinear variable = lagged public debt
	Threshold variable = lagged public debt
		All	Advanced	16 Emerging	21 Emerging
		countries	countries	countries	countries
		1790-2009
	Test of nonlinearity		bootstrapped p-value
	H0: linear vs. H1: 2 regimes	0.064	0.018	0.134	0.218
	H0: 2 regimes vs. H1: 3 regimes	0.218	0.010	0.146	0.226
	Coefficients
	Low debt	0.011	0.061 **
	Middle debt	-0.001	0.015 *
	High debt		-0.001
	Debt thresholds (%)
	Threshold 1	40.51	20.38
	Threshold 2		55.35
	Country fixed effects	YES	YES	YES	YES
	No. obs	4700	2881	1634	1880
		1946-2009
	Test of nonlinearity		bootstrapped p-value
	H0: linear vs. H1: 2 regimes	0.012	0.000	0.060	0.050
	H0: 2 regimes vs. H1: 3 regimes	0.116	0.104	0.304	0.376
	Coefficients
	Low debt	0.047 ** 0.063 **	-0.021	-0.032 *
	Middle debt
	High debt	-0.004	-0.008 *	0.004	-0.002
	Debt thresholds (%)
	Threshold 1	16.42	20.38	38.02	36.10
	Country fixed effects	YES	YES	YES	YES
	No. obs	2220	1236	896	1037
	Note: * and ** denote statistical significance at the 10% and 5% levels, respectively.

Table 3 . The nonlinear relation between public debt and growth, annual data Endogenous debt thresholds

 3

	Nonlinear variable = lagged rate of growth of public debt
	Threshold variable = lagged public debt
		All	Advanced	16 Emerging	21 Emerging
		countries	countries	countries	countries
		1790-2009
	Test of nonlinearity	bootstrapped p-value
	H0: linear vs. H1: 2 regimes	0.024	0.076	0.034	0.030
	H0: 2 regimes vs. H1: 3 regimes	0.030	0.338	0.054	0.036
	Coefficients
	Low debt	0.000	-0.002	0.001	0.000
	Middle debt	-0.024 **		-0.026 **	-0.023 **
	High debt	-0.004	-0.036 *	0.000	0.002
	Debt thresholds (%)
	Threshold 1	28.90	73.73	17.45	17.39
	Threshold 2	51.72		54.18	50.99
		1946-2009
	Test of nonlinearity	bootstrapped p-value
	H0: linear vs. H1: 2 regimes	0.004	0.104	0.000	0.004
	H0: 2 regimes vs. H1: 3 regimes	0.000	0.024	0.028	0.028
	Coefficients
	Low debt	-0.001		-0.001	-0.001
	Middle debt	-0.033 **		-0.045 **	-0.036 **
	High debt	-0.002		-0.010	-0.007
	Debt thresholds (%)
	Threshold 1	18.39		18.46	17.71
	Threshold 2	60.83		50.57	53.84
	Note: * and ** denote statistical significance at the 10% and 5% levels, respectively.

Table 4 . Advanced OECD countries, 1946-2009, non-overlapping multi-year averages

 4

		Nonlinear variable = public debt/GDP	Nonlinear variable = growth rate of public debt/GDP
		Threshold variable = public debt/GDP	Threshold variable = public debt/GDP
		5-year	8-year	10-year	5-year	8-year	10-year
	Test of nonlinearity			bootstrapped p-value
	H0: linear vs. H1: 2
	regimes	0.004	0.024	0.004	0.044	0.018	0.006
	H0: 2 regimes vs. H1: 3
	regimes	0.002	0.106	0.012	0.086	0.106	0.134
	Coefficients
	Low debt	0.098 ** -0.034 **	0.11 **	0.018	-0.005	-0.068 **
	Middle debt	0.034 *		0.031	-0.064 **
	High debt	-0.005	-0.017 **	-0.01	-0.008	-0.095 **	-0.289 **
	Debt thresholds (%)
	Threshold 1	18.61	58.17	18.41	17.47	42.56	67.04
	Threshold 2	33.27		36.30	39.13
	Country fixed effects	YES	YES	YES	YES	YES	YES
	No. obs	238	140	140	237	139	139
	No. countries	20	20	20	20	20	20
	Note: * and ** denote statistical significance at the 10% and 5% levels, respectively.

Table 5 . The nonlinear effect of public debt on growth Polynomial functional form of nonlinearity

 5

		All	Advanced	16 Emerging	21 Emerging
	countries		countries	countries	countries
	1790-2009 -0.026** 0.00008** -0.027** 0.00009** -0.001	-0.00014	-0.013	-0.00003
	1790-1936 -0.015	0.00005	-0.014	0.00006	0.051	-0.00046** 0.014	-0.00016
	1946-2009 -0.070** 0.00038*	-0.076** 0.00042*	-0.044** 0.00017	-0.048** 0.00019
	1790-2009 -0.011*	0.00004	-0.019** 0.00006*	-0.010	0.00014	-0.003	0.00006
	1790-1936 0.013	-0.00002	0.005	0.00000	0.026	0.00000	0.026	-0.00003
	1946-2009 -0.025** 0.00011** -0.039** 0.00015** -0.032*	0.00037**	-0.032*	0.00032**
	Note: * and ** denote statistical significance at the 10% and 5% levels, respectively. The estimations are carried out
	with country fixed effects.

Table 4a . NONLINEAR MODEL AVERAGING General government debt and growth 29 OECD countries, 1960-2010

 4a

		Nonlinear variable = lagged public debt
		Threshold variable = lagged public debt
		5-year averages	8-year averages	10-year averages
	Minimum % of observations in one regime 10% 20% 30% 10% 20% 30% 10% 20% 30% 10-2 10-3 posterior inclusion probability
	Linear regime 0.009 0.016 0.052 0.071 0.002 0.001 0.000 0.000 0.000
	2 regimes	0.003 0.284 0.344 0.158 0.457 0.416 0.076 0.003 0.002
	3 regimes	0.790 0.700 0.605 0.077 0.541 0.583 0.911 0.997 0.998
	posterior mean conditional on inclusion
	Low debt	-0.041 -0.017 -0.019	--	0.010 0.008 -0.034 0.000 0.000
	Middle debt -0.011 -0.002 -0.004	--	0.031 0.029 -0.011 0.013 0.012
	High debt	-0.021 -0.008 -0.010	--	-0.014 -0.015 -0.021 0.066 0.065
	debt thresholds (%)
	Threshold1	27.70 24.54 21.20	--	23.21 25.00 36.95 42.14 41.93
	Threshold2	53.02 46.98 40.58	--	34.02	36.63 64.53 70.64 70.72
	Note: bold figures indicate that the estimated posterior inclusion probability is higher than 0.50

Table 4d . NONLINEAR MODEL AVERAGING The rate of growth of general government debt and growth 13 OECD countries, 1960-2010

 4d

	Nonlinear variable = lagged growth rate of public debt
		Threshold variable = lagged public debt
		5-year averages	8-year averages	10-year averages
	Minimum % of observations required in one regime 10-2 10% 20% 30% 10% 20% 30% 10% 20% 30% 10-3 posterior inclusion probability
	Linear	0.000 0.035 0.035 0.000 0.004 0.004 0.015 0.134 0.189
	2 regimes	0.961 0.965 0.965 0.016 0.157 0.154 0.190 0.772 0.737
	3 regimes	0.000 0.000 0.000 0.941 0.839 0.843 0.669 0.094 0.074
	posterior mean conditional on inclusion
	Low debt	0.010 0.008 0.008 0.008 -0.004 -0.004 0.026 0.025 0.049
	Middle debt		0.071 0.042 0.043 0.057
	High debt	0.080 0.079 0.079 -0.082 -0.070 -0.070 0.009 0.068 0.060
	debt thresholds (%)
	Threshold1 42.05 43.14 43.14 40.49 36.62 36.76 27.42 53.51 45.86
	Threshold2		57.13 50.93 51.14 47.02
	Note: bold figures indicate that the estimated posterior inclusion probability is higher than 0.50

Table A1 . The nonlinear relation between lagged public debt and growth, 1790-2009, annual data EXOGENOUSLY IMPOSED DEBT THRESHOLDS (30%, 60%, 90% of government debt)

 A1

		ALL COUNTRIES		DEVELOPED COUNTRIES	16 EMERGING COUNTRIES		21 EMERGING COUNTRIES
	1790-	1790-	1946-	1790-	1790-	1946-	1790-	1790-	1946-	1790-	1790-	1946-
	2009	1939	2009	2009	1939	2009	2009	1939	2009	2009	1939	2009

Nonlinear variable = lagged central government debt; Threshold variable = lagged central government debt

	Linear model	-0.003	0.008	*	-0.007	**	-0.006	**	0.005		-0.013	**	0.009		0.026	*	0.010		0.007		0.020	*	0.004
	2-regime model
	debt<90%	0.005	0.031	**	-0.004		0.000		0.026	**	-0.016	**	0.017	**	0.046	**	0.014	*	0.014	**	0.042	**	0.008
	debt>=90%	-0.003	0.011	**	-0.008	**	-0.005	*	0.009		-0.013	**	0.006		0.023		0.003		0.005		0.020	*	0.000
	3-regime model
	debt<=60%	-0.003	0.018		-0.017	**	-0.001		0.018		-0.029	**	-0.004		0.025		0.002		-0.003		0.029	0.000
	60%<=debt <90%	-0.005	0.014		-0.013	**	-0.007		0.011		-0.022	**	0.002		0.027		0.005		0.000		0.025	-0.001
	debt>=90%	-0.003	0.009	*	-0.007	**	-0.005	*	0.006		-0.014	**	0.010		0.026		0.013	*	0.007		0.021	*	0.008
	4-regime model
	debt<30%	-0.001	0.009		-0.016		0.024		0.039		-0.015		-0.027		-0.028		0.003		-0.023		-0.021	-0.010
	30%<=debt <60%	-0.003	0.017		-0.017	**	0.004		0.021		-0.026	**	-0.009		0.013		0.002		-0.008		0.018	-0.002
	60%<=debt <90%	-0.005	0.013		-0.013	**	-0.003		0.013		-0.020	**	-0.002		0.016		0.005		-0.003		0.016	-0.003
	debt>=90%	-0.003	0.009	*	-0.007	*	-0.003		0.007		-0.012	**	0.007		0.019		0.013		0.005		0.016	0.007
	Nonlinear variable =

lagged rate of growth of central government debt; Threshold variable = lagged central government debt

  : * and ** denote statistical significance at the 10% and 5% levels, respectively. The estimations are carried out with country fixed effects.

	Linear model	-0.002		-0.002		-0.003		-0.004		-0.001	-0.004		-0.001		0.006		-0.003		-0.001		-0.004	-0.002
	2-regime model
	debt<90%	-0.001		0.000		-0.003		-0.002		-0.001	-0.004		-0.001		0.002		-0.003		-0.001		0.003	-0.002
	debt>=90%	-0.005		0.006		-0.001		-0.030		-0.013	-0.026		0.014		0.022		0.005		0.015		0.022	0.002
	3-regime model
	debt<=60%	-0.002		-0.002		-0.003		-0.002		0.000	-0.010		-0.001		0.006		-0.002		-0.002		-0.007	-0.002
	60%<=debt <90%	-0.007		-0.011	*	-0.002		-0.003		-0.006	-0.001		-0.022	**	-0.023	**	-0.018		-0.016	**	-0.021	**	-0.009
	debt>=90%	0.001		0.014		0.000		-0.031		-0.010	-0.082	**	0.024		0.038		0.008		0.023		0.039	0.005
	4-regime model
	debt<30%	0.000		0.000		-0.001		-0.001		0.000	0.000		0.000		-0.001		-0.001		0.000		-0.002	-0.002
	30%<=debt <60%	-0.020	**	-0.011		-0.033	**	-0.018	*	-0.014	-0.027	**	-0.018		0.015		-0.041	**	-0.020	**	-0.010	-0.035	**
	60%<=debt <90%	-0.007		-0.011	*	-0.002		-0.003		-0.006	-0.001		-0.022	**	-0.023	**	-0.018		-0.016	**	-0.021	**	-0.009
	debt>=90%	0.001		0.014		-0.001		-0.031		-0.010	-0.082	**	0.023		0.039		0.008		0.023		0.039	0.004

Note

Table A4 . Advanced OECD countries, 1946-2009, non-overlapping multiyear averages, sensitivity check

 A4

	Minimum % of observations required in one regime
	30%	10%	5%
	Nonlinear variable =

level of the public debt to GDP ratio Threshold variable= public debt to GDP ratio

							10-
		5-year	8-year	10-year	5-year	8-year	year	5-year	8-year	10-year
	Test of nonlinearity					bootstrapped p-value
	H0: linear vs. H1: 2 regimes	0.002	0.048	0.010	0.002	0.024	0.004	0.002	0.020	0.008
	H0: 2 regimes vs. H1: 3
	regimes	0.246	0.252	0.006	0.002	0.004	0.002	0.004	0.008	0.006
	Coefficients
	Low debt	0.032 *	0.028	-0.003	0.186 ** 0.226 **	0.189 ** 0.186 ** 0.226 **	0.189 **
	Middle debt		-0.009	-0.04 ** 0.046 ** 0.047 **	0.041 *	0.046 ** 0.047 **	0.041 *
	High debt	-0.01 *		-0.022 ** -0.004	-0.003	-0.009	-0.004	-0.003	-0.009
	Debt thresholds (%)
	Threshold 1	32.47	31.56	36.30	12.51	11.98	12.30	12.51	11.98	12.30
	Threshold 2			54.53	33.27	31.56	36.30	33.27	31.56	36.30
		Nonlinear variable =

rate of growth of the public debt to GDP ratio Threshold variable= public debt to GDP ratio

	Figure A1.
							10-
		5-year	8-year	10-year	5-year	8-year	year	5-year	8-year	10-year
	Test of nonlinearity				bootstrapped p-value
	H0: linear vs. H1: 2 regimes	0.022	0.034	0.018	0.022	0.016	0.020	0.044	0.024	0.010
	H0: 2 regimes vs. H1: 3
	regimes	0.282	0.374	0.210	0.088	0.100	0.074	0.090	0.088	0.096
	Coefficients
	Low debt	-0.009	-0.005	-0.037	0.018	-0.005	-0.018	0.018	-0.006	-0.018
	Middle debt				-0.064 **		-0.099 ** -0.064 ** -0.149 **	-0.099 **
	High debt	-0.078 ** -0.095 **	-0.151 ** -0.008	-0.095 **	-0.298 ** -0.008	-0.025	-0.298 **
	Debt thresholds (%)
	Threshold 1	52.62	42.56	26.08	17.47	42.56	15.36	17.47	42.56	15.36
	Threshold 2				39.13		67.04	39.13	61.98	67.04

Annual real GDP growth and lagged central government debt as a % of GDP

		Panel A. All countries
	1790-2009	1790-1939	1946-2009
		Panel B. Developed economies
	1790-2009	1790-1939	1946-2009
		Panel C. 21 emerging market economies
	1790-2009	1790-1939	1946-2009
	Source: Author's calculations

But we still use the debt-to-GDP ratio as the threshold variable.

 [START_REF] Reinhart | Growth in a time of debt[END_REF]Our dataset, which draws on [START_REF] Reinhart | From financial crash to debt crisis[END_REF] for the level of central government debt [START_REF] Barro | Barro Ursúa macroeconomic dataset[END_REF] 1969-20091969-2009South Africa 1950-20091912-2009Sri Lanka 1950-20091871-2009Thailand 1950-20091987-2009Turkey 1933-20091875-2009Uruguay 1935-20091971-2009Venezuela 1921-20091914-2009


EconomiX Christian Bidard
email: christian.bidard@u-paris10.fr.

THE FRAIL GROUNDS OF THE RICARDIAN DYNAMICS

Keywords: Rent, Ricardian dynamics, Ricardo, Sraffa JEL classification. B12, C61, D30

The Ricardian dynamics study the evolution of distribution when demand increases. The successive marginal methods in agriculture are determined by means of a rule which has never been stated explicitly by Ricardo's commentators.

But the rule also allows us to point at two limits of Ricardo's construction, first in his attempt to get rid of rent in the analysis of distribution, second in the working of the dynamics themselves. Similarly, the identification of a productivity and a profitability criterion is at the basis of Sraffa's mistake in his reconstruction of the theory of rent, whereas post-Sraffian formalizations have abandoned Ricardo's dynamic approach.

The Ricardian dynamics are among the most classical pieces of economic analysis: Ricardo (1817) combined parsimonious assumptions with a deep analysis of the extension of cultivation, from which he drew dramatic conclusions on the evolution of distribution between rents and profits and a plea for free trade. Consider three classes in society: the workers, who earn wages, the landowners, who rent lands to farmers, and the capitalists, who invest capital in industry (or, in the case of farmers, in agriculture), pay the workers and receive profits on their advances. It is assumed that transfers of capital from less to more profitable sectors lead to the uniformity of the rates of profits across sectors. Ricardo examined the impact of the long-run dynamics of capitalism on distribution. The real wage may fluctuate in the short run depending on economic conjuncture, but in the long run the Malthus law exerts a strong pressure against a significant increase above a socially defined minimum subsistence level: we may assume that the long-run real wage is constant and represented by a basket of commodities, with bread as its main component.

Therefore, the development of industry and the employment of more workers require the production of more and more wheat and, in a closed economy, poorer and poorer lands are cultivated. In contrast with the extension of cultivation on a given quality of land, which has no incidence on values, the cultivation of a land of a lower grade requires a rise in the price of corn, that must be sufficient to make the investment on the rather bad quality of land as profitable as in any other sector of the economy. But, then, the owners of the more fertile lands, on which the production costs are lower, are in a position to demand a rent of their farmers: the level of the rent compensates for the gap between the production costs on the marginal land, the last to be cultivated, and those on the intramarginal lands of better qualities which were cultivated before.

As the marginal methods pay no rent, the prices are determined by the conditions of production in industry and on the marginal lands. That 'separation property' allows Ricardo to discard rent from the analysis of value, as if there were no scarce resources. Then the trade-off property between wages and profits, a basic law of distribution established in the absence of lands, also holds in their presence. In the long run, the rents surge because of the increasing gap between the worst and the best lands. Constant wages and increasing rents lead to a squeeze of profits: the uniform rate of profits falls, there is no more incentive to invest and the economy reaches a steady state. An alternative to the cultivation of new lands may be to let more capitalistic, also more productive, agricultural methods operate on the best lands. Taking that possibility into account does not change the conclusions, however, as the principle according to which "the capital last employed pays no rent" (Principles, Chapter II) applies to intensive as well as to extensive cultivation.

It is often admitted that, under the retained hypotheses, which ignore technical progress, Ricardo's description of the dynamics is correct and general. Our aim is to study their analytical grounds in more details. Apart [START_REF] Ricardo | On the Principles of Political Economy and Taxation[END_REF], we shall also refer to [START_REF] Sraffa | Production of Commodities by Means of Commodities[END_REF] who developed an explicit formalization of prices and rents, and to post-Sraffian economists who discovered a certain number of unexpected phenomena in rent theory. Useful as they may be, most of these modern studies miss the central point of Ricardo's construction, viz. the dynamics themselves, and do not examine the law of succession of marginal methods when demand increases.

We thus start from Ricardo and explain the nature of that law, which is governed by the 'minimum rule' (Section 1). Ricardo did not see, however, some logical consequences of the rule. We examine two of them: first, the working of the dynamics requires a condition to guarantee that the physical and the value sides of an equilibrium both pull in the same direction (Section 2). Second, from a methodological point of view, the validity of Ricardo's attempt to get rid of rent in the analysis of value and distribution is dubious (Section 3).

The analysis holds independently of the distribution variable chosen as exogenous, be it the real wage (Ricardo) or the uniform rate of profits (Sraffa). For reasons which will be explained, calculations are much simpler under the given rate of profits hypothesis. Even if this departure from Ricardo's own hypothesis is unfortunate from a historical perspective, we first retain that assumption, then check that the phenomena we point at hold under the alternative hypothesis (Section 4).

Next, we apply the same device to Sraffa's approach and put into light the source of Sraffa's mistake in his analysis of production with land, then conclude with a brief survey of post-Sraffian studies on rent (Section 5).

The main points are illustrated by means of numerical examples in onecommodity and, in the Appendix, a two-commodity model. Clearly enough, the apparent paradoxes with regard to Ricardo's construction which occur in simple models also hold in general multisector models.

THE MINIMUM RULE

When the present marginal land becomes fully cultivated, cultivation can be extended on a new land provided that the price of corn increases sufficiently to justify that investment. Which new land, and by means of which new method? For Ricardo, a given grade of land is characterized by its physical qualities, which are described by the amounts of physical inputs, land and labor needed for the production of corn.

Ricardo classified the lands 1, 2, 3, etc., according to their grade. The order of cultivation is defined a priori, but that classification relies on the costs of production on the various lands, normal profits included. Ricardo's note attached to the last word of Chapter II shows the way to calculate the price of corn when its cultivation is extended to a land of lower quality. The price of corn when land 2 is marginal is higher than when land 1 is marginal, and it is clear from Ricardo's example that cultivation has been extended to land 2 rather than to land 3 because the price of corn is smaller on land 2. In other words, when land 1 has become fully cultivated, the price of corn rises up to the minimum level which allows for the extension of cultivation on another land. Less immediate and more instructive is the choice between an extension of cultivation on another land or its intensification on the same land, because the issue is uncertain. Ricardo gave the following hint: "It often, and, indeed, commonly happens, that before No. 2, 3, 4, or 5, or the inferior lands are cultivated, capital can be employed more productively on those lands which are already in cultivation" (Principles, Chapter II). 'Before' has a chronological, also a logical meaning: the cultivation of other lands would certainly become profitable for a sufficient rise in the price of corn but, in a given situation, the rise stops at the first value making some new agricultural method profitable, given the adjustment in rent.

Indeed, if the rise went beyond that minimum level, the profitability of the corresponding method would continue to increase and would exceed the ruling rate of profits. That is what we dub the minimum rule.

A numerical example in a one-commodity model illustrates the rule and the dynamics. Let corn be initially cultivated on land 1 by means of method (1): qr. corn + 1 d. labor + 1 acre land₁ → 1 qr. corn

(1) (the physical units called 'quarter', 'day' and 'acre' are conventional). Assume that the given uniform rate of profits is r = 10% (that rate will be maintained in all examples).

The rent is zero as long as land is not fully cultivated and, for a wage paid post factum (Sraffa's hypothesis) and set equal to one, the price p of corn is equal to 2.

Before considering any alternative method, we know that, when land 1 will be fully cultivated, the price of corn will rise from 2 to 2 + h (h > 0) and that the landowner will get a positive rent per acre ρ = ρ(h): "A rent is paid because corn is high" (Principles,

Chapter II). The price-and-rent equation associated with method ( 1)

(1 + 10%) (2 + h) + 1 + ρ = 2 + h
shows that ρ = 0.5h. Suppose now that the increasing demand can be met either by operating a more intensive method of cultivation (2) on land 1 (more inputs and more production per acre, and a positive rent):

qr. corn + 3 d. labor + 1 acre land₁ → 1.6 qr. corn (2)

or by extending the cultivation to land 2 (more inputs and same production par acre, but a zero rent) by method (3):

qr. corn + 1 d. labor + 1 acre land2 → 1 qr. corn (3)
These methods are not profitable at the initial price p = 2 but their profitability improves with the price of corn, therefore with the level of h. The level of h₂ of h for which method (2) becomes profitable is the solution of

(1 + 10%) (2 + h) + 3 + 0.5h = 2 + h hence h₂ = 2.
A similar calculation shows that the extensive method becomes profitable at h₃ = 3. Therefore, the price of corn rises from p = 2 to p′ = 2 + h₂ = 4 and the intensive method on land 1 is operated before an extension to land 2.

In Chapter II of the Principles, Ricardo adopts the farmers' point of view and sets forth the minimum rule in terms of price as a maximum rule in terms of product: a given amount of capital is invested where the product is maximum. Whatever its formulation, the rule is a specific application of the general law of the one and minimum price induced by competition. This does not explain why the interactions of the rule and the dynamics are hardly mentioned in the economic literature on Ricardo and why their consequences have been ignored.

FAILURE OF THE DYNAMICS

A long-term equilibrium satisfies two types of constraints: on the quantity side, the operated methods must meet the given level of final demand; on the value side, they must yield the ruling rate of profits. The basic remark for the study of the dynamics is:

starting from a given equilibrium, its limit is determined by the quantity side, whereas the reconstruction of a new equilibrium as defined by the minimum rule starts from the value side. Ricardo failed to notice some dramatic effects of that double determination.

Let the demand for corn increase and suppose that the present equilibrium reaches its physical limit. If the incoming method consists in extending the cultivation of corn on a new land, the productive system does meet the evolution of demand.

Otherwise, the issue is unclear. To illustrate the point, let us return to the above corn model with methods (1)-( 2)-( 3) and introduce another method of cultivation on land 1: 0.1 qr. corn + 1.03 d. labor + 1 acre land₁ → 0.62 qr. corn (4)

The same calculations as above show that method (4) becomes profitable at h₄ = 1, before methods (2) or (3): the price of corn cannot rise further, otherwise method (4) would pay extra profits. But the net productivity of that method (0.52 qr. per acre) is smaller than that of method (1) it replaces progressively (6/11 qr., i.e. about 0.54 qr.

per acre): the introduction of the new method reduces the net product and does not help to meet a higher demand. The reason of the phenomenon is clear: the choice of the incoming method is based on its profitability, not on its productivity, and a contradiction here appears between the value and the quantity side of the equilibrium problem. In that case, the Ricardian dynamics fail.

Two comments are in order. First, it might be shown that the dynamics always work when the rate of profits is zero, because of the duality property which then holds between the quantity side and the value side of the model (golden rule). Second, a demand corresponding to an intermediate productivity, say 0.53 quarters per acre, can be met by two distinct equilibria: one consists of method (1) used alone on 53/54th of land 1, the other of the joint use of methods (1) and ( 4), each on half on the soil. Each of these equilibria is sustained by a specific price-and-rent vector (p c = 2, ρ = 0 for the first equilibrium, p c = 2 + h₄ = 3, ρ = 0.5h₄ = 0.5 for the second): the failure of the dynamics at some level of demand gives birth to a multiplicity of equilibria at a slightly smaller level.

FAILURE OF THE METHOD

We now assume that the dynamics work, so that the critique developed here is independent of that of Section 2.

The minimum rule also applies to multisector models, but its above statement is incomplete. In Chapter II of the Principles Ricardo stressed the effects of a rise in the price of corn on rents, but he was also aware that this rise also triggers other price changes. The phenomenon is analyzed at length in the chapters of the Principles devoted to the effects of taxation ("By raising the price of raw produce, the prices of all commodities into which raw produce enters would be raised", Chapter IX). Such rises, with changes in relative prices, are necessary to maintain the uniformity of the rates of profits among the operated methods. We shall refer to them as the 'compensating variations' in prices and rents. Their calculation obeys the same principles as for the change in rents (an example is given in the Appendix) and, following an increase h in the price of corn, leads to the determination of an increasing price p i = p i (h) for commodity i alongside with increasing rent(s) ρ = ρ(h).

The exact statement of the minimum rule is that the new operated method is the first which becomes profitable when the price of corn increases, the compensating variations for rents and prices being taken into account.

What are the consequences of an increase of the price of corn on the profitability of an industrial method, say in a corn-iron model in which each commodity enters into the production of the other? By construction, the compensating variations are such that the presently operated iron method maintains its rate of profits. As for a non-operated iron method, two effects work in opposite directions: the cost of production increases, the value of the product too, therefore the evolution of the rate of profits is unclear. But suppose that the new iron method I₁ we consider is corn saving with regard to the presently operated method I₀. As the weight of corn in the costs of I₁ is smaller than for method I₀, the second effect dominates and the profitability of the iron method I₁ increases with h. Depending on numerical data, method I₁ may reach the normal rate of profits before any agricultural method and is the incoming method. Then the new long-term equilibrium is characterized by the coexistence of two iron methods and the dynamics see the progressive substitution of the corn-saving method I₁ for I₀, therefore the economy does produce more corn. The phenomenon at stake is similar to that of intensive cultivation, but the substitution of methods here takes place in industry instead of agriculture.

It is an essential feature of Ricardo's approach to get rid of rent by considering the conditions of production of marginal capital, which pays no rent. It is that separation property which explains the architecture of the Principles: Chapter I ("On

Value") is concerned by the determination of values and prices of the reproducible commodities. The main exception does not concern "wines of a peculiar quality, which can be made only from grapes on a particular soil", but land, which is not produced by men. In Chapter II ("On Rent"), Ricardo reduces the study a productive system with land to another without land, made of the industrial methods and the marginal agricultural methods. The properties of single-product systems (which, more than one century later, will be proved formally by means of the Perron-Frobenius theorem) then apply. Wages and profits are determined in a simple productive system, as if there was no scarce resource. In a second time, rents are reintroduced and determined as differential costs.

Is the separation property universal? The long-term equilibrium we consider admits two operated iron methods whereas land is fully cultivated by means of a unique agricultural method. A similar two-step procedure applies: the prices of corn and iron are first determined, then the level of rent is obtained by comparing the production costs and the price of corn. However, in the first step of the procedure, the price equations are those associated with two methods producing the same good, so that the standard properties of single-product systems do not hold. In particular, the trade-off property between wages and profits cannot be expected, as illustrated in the Appendix. We dubbed 'non-Ricardian' an equilibrium of this type, with no marginal agricultural method. Ricardo's analytical strategy fails in the unavoidable presence of non-Ricardian equilibria.

The distinction between Ricardian and non-Ricardian equilibria can be defined precisely. Let an agricultural method be called marginal if it is operated only on a part of a given quality of soil. That soil itself may be cultivated either partly (extensive cultivation) or totally (intensive cultivation), in which case at least two methods are used jointly. Then the land itself is called marginal. By contrast, the intramarginal lands are totally cultivated by means of a unique method. For any type of equilibrium, be it Ricardian or non-Ricardian, the equality card(operated methods) = card(commodities) + card(fully cultivated lands) (card is for cardinal and indicates the number of elements of a set) holds flukes apart.

That equality explains why, for a given rate of profits, the prices and rents are uniquely determined when the operated methods are known. For a Ricardian equilibrium, the equality splits into two:

card(marginal methods) = card(agricultural goods) + card(marginal lands) card(operated industrial methods) = card(industrial goods)

Are non-Ricardian equilibria exceptional? From a logical point of view, one example suffices to criticize Ricardo's methodology. From a formal point of view, the above splitting becomes unlikely when the numbers of goods and lands are great: there are strong reasons to think that non-Ricardian equilibria are more frequent than Ricardian equilibria in multisector complex economies. With multiple agricultural goods, Ricardian equilibria are even more improbable.

GIVEN REAL WAGE

Up to now, it has been assumed that the rate of profits is the independent distribution variable, but the conclusions hold under Ricardo's given real wage hypothesis. For a wage defined in physical terms as a basket (δ₁,..., δ n ) of commodities, the additional equality δ₁p₁+...+ δ n p n = 1 is introduced and the rate of profits becomes an endogenous magnitude. From a theoretical point of view, the compensating variations can be determined as well. One can first replace each unit of labor by the wage basket, and then labor disappears from the model with augmented input coefficients. The factor of profits is the inverse of dominant root of the extended input matrix and the initial price vector is the associated Perron-Frobenius vector. When corn becomes scarce, its price increases by some scalar h and the compensating variations concern rents and the other prices. This leads to relationships of the type r = r(h), ρ = ρ(h) and p = p(h). For h increasing up from zero, the incoming method is the first among those which are not initially operated whose profitability at prices p = p(h) and rents ρ(h) becomes equal to r(h). In multisector models, explicit calculations are rapidly overwhelming because they deal with Perron-Frobenius characteristics depending on a parameter. However, as far as the objective is only to exhibit either studied in Section 2, method (2) is more costly when method (1) operates alone, but less costly when they operate simultaneously. Sraffa, who was aware of the pitfalls of capital theory, seems to have forgotten them at that point: more capital does not mean more product. The only surprise, if any, is that the problem occurs even in a simple corn model with land.

The post-Sraffian studies on rent started with Quadrio-Curzio (1966). They were initially devoted to the analysis of simple cases, with a unique agricultural product: [START_REF] Montani | Scarce natural resources and income distribution[END_REF] showed that, for a given rate of profits and extensive cultivation proper, the order of cultivation coincides with that of the decreasing wages paid by the agricultural methods for a zero rent, a property in line with Ricardo's dynamic approach. (Note that the definition of extensive cultivation retained in post-Sraffian studies includes the unexpected hypothesis that the industrial methods are given. The hidden reason of that restriction is that non-Ricardian equilibria are thus discarded.)

In contrast with Ricardo, Sraffa did not refer explicitly to demand (he mentioned the short supply of lands as the cause of a change of methods) but introduced the notion of 'requirements for use'. A given demand vector was later explicitly introduced, first in the formalization of the notion of equilibrium for multipleproduct systems [START_REF] Steedman | Positive profits with negative surplus value: a reply to Mr[END_REF][START_REF] Schefold | On counting equations[END_REF], then to deal with the scarcity constraints on lands. Either under the influence of Sraffa's preface or of that formalization, Ricardo's dynamic approach was abandoned and most post-Sraffian papers on rent dealt with the solutions of a system of equations (or, equivalently, of inequalities with complementarity relationships) for a given demand basket. These equations do include a cost-minimization property, but the minimum rule and Ricardo's idea that a small increase in demand is dealt with by introducing one new marginal method are ignored. A partial exception is [START_REF] Saucier | Le choix de techniques en situation de limitation des ressources[END_REF], who explained why the net product of corn may increase thanks to a substitution of method in industry. He dubbed that phenomenon, which we call non-Ricardian equilibrium, 'external differential rent'. Beyond the post-Sraffian stream of thought, the same static approach, characterized by a symmetric treatment of the quantity side and the value side, is adopted by all studies of Ricardo's works, even those which, like [START_REF] Morishima | Ricardo's Economics. A General Equilibrium Theory of Distribution and Growth[END_REF], are analytically oriented.

a failure of the dynamics themselves. A century and half later, Sraffa, in spite of some critiques, approved most of Ricardo's conclusions. A return to the dynamics also explains the apparent paradoxes discovered by post-Sraffians in a static framework.

That case exemplifies an instance in which contemporary analytical studies would have benefited from a closer reading of the original works.

APPENDIX

The Appendix illustrates the statements of Section 3 in a bisector model. The rate of profits is r = 10%, wages are paid post factum and labor is chosen as numeraire.

Compensating variations

Let there be a corn-iron model in which the present marginal agricultural method ( 5) and the operated industrial method ( 6

As long as land 1 is not fully cultivated, the rent is zero and the prices in terms of wage are p c = 1 for corn and p i = 1 for iron. When land 1 becomes fully cultivated, let the price of corn rise up to p c = 1 + h for some positive scalar h considered as a parameter. The rent ρ becomes positive and the price of iron rises. The relationships between these values are given by Sraffa's price equations

(1 + 10%)( p c + p i ) + 0.4 + ρ = p c

(1 + 10%)( p c + p i ) + 0.1 = p i from which there follows that the ruling rate of profits is maintained for both methods if p c = 1 + h, p i = 1 + 0.8h, ρ = 0.46h (7)

Determination of the incoming method

Let a (potential) intensification method (8) on land 1 and a (potential) extension method (9) on land 2 be respectively described as 2 qr. corn + 1 t. iron + 1 d. labor + 1 acre land₁ → 4 qr. corn (8) 0.3 qr. corn + 0.3 t. iron + 1 d. labor + 1 acre land₂ → 1 qr. corn (9)

These methods yield less than the ruling rate of profits at the prices associated with the initial equilibrium, but their profitability improves with the rise in the price of corn.

For prices and rent given by relations (7), the intensive method reaches the normal rate of profits for h₈= 0.65 and the extensive method for h₉= 1.62 > h₈. The minimum rule leads to the introduction of the intensive method.

A non-Ricardian equilibrium

Let us moreover introduce another iron method (10):

qr. corn + t. iron + 0.4 d. labor → 1 qr. iron (10)

The input coefficients show that method (10) is corn saving with regard to method (6).

For prices given by ( 7), its profitability improves with h and reaches the ruling rate of profits at h₁₀ = 0.5, before methods ( 8) and ( 9). The minimum rule leads to the introduction of method (10). The prices are then determined by both iron methods therefore the real wage in terms of corn and the rate of profits vary in the same direction.

  At r = 10%, the solution of these equations are p c = 1 + h₁₀ = 1.5 and p i = 1 + 0.8h₁₀ = 1.4. But, for a moving rate of profits, it turns out that the price of corn in terms of wage is a decreasing function of r p c = 1.5.

the failure of the dynamics or non-Ricardian equilibria, numerical examples are easily built.

It turns out that the main technical advantage of Sraffa's hypothesis is that the price equations are linear when the rate of profits is given, with simple formulas for the compensating variations. By contrast, the same system is nonlinear when the unknowns are the prices and the rate of profits. The gap does not affect the economic conclusions.

MODERN ANALYSES

The modern revival of Ricardian studies is due to Sraffa (Production, 1960). Following Ricardo's approach, Sraffa first introduced a general formalization of prices of production and studied their properties (Chapters I to VI of Production), then considered lands and rents (Chapter XI) as part of a general theory of multipleproduct industries. It is worth noting that, in spite of the explicit warning of the preface that "no changes in output" are ever considered, Sraffa followed Ricardo's dynamic approach when he studied the effects of "a progressive increase of production" (Section 88). He rejected Ricardo's idea of fertility as an intrinsic quality of lands determining a natural order of cultivation, because the decision to cultivate some land rather than another depends on relative costs, which are sensitive to the distribution of income. Sraffa studied the cases of extensive and intensive cultivation proper and was optimistic as to the possibility to extend the results to several agricultural goods (Section 89).

In his analysis of intensive cultivation, Sraffa specified that both methods are operating side by side on the same land. What are the economic conditions sustaining that coexistence? The price of corn is unique and both methods must be equally profitable. Let the cost per quarter produced be defined as including the cost of material inputs, wages and profits, but excluding rent. Then "the method that produces more corn per acre [shows] a higher cost" (Section 87) and "a progressive increase of production on the land [...] takes place through the gradual extension of the method that produces more corn at a higher unit cost, at the expense of the method that produces less" (Section 88). But the conjunction of both statements does not provide sufficient grounds to the Ricardian dynamics, since the cost order between methods depends on prices and is not an invariant: in the corn model Post-Sraffian contributions led to a partial rejection of Sraffa's optimistic conclusions and put forward a number of unexpected phenomena and difficulties. [START_REF] Salvadori | Land and choice of techniques within the Sraffa framework[END_REF] established a general existence result for a given demand vector, but D'Agata (1982; see also [START_REF] Freni | Capitale tecnico nei modelli dinamici ricardiani[END_REF] provided examples of multiple equilibria.

Starting from geometrical considerations, [START_REF] Erreygers | On the uniqueness of square cost-minimizing techniques[END_REF][START_REF] Erreygers | On the uniqueness of square cost-minimizing techniques[END_REF] elaborated an algebraic criterion for local uniqueness which compares the sign of two determinants.

As noticed as the end of Section 2, the failure of the dynamics gives rise to a local multiplicity of equilibria, and therefore another interpretation of the Erreygers criterion is that it ensures the local working of the Ricardian dynamics. In a study of intensive rent proper for a given real wage, [START_REF] Bidard | The dynamics of intensive cultivation[END_REF] returned to the dynamic approach by reintroducing the idea that the incoming method is commanded by the value side. Bidard (2012) provides an analytical study of the dynamics and draws a close parallel between Ricardo's method and the [START_REF] Lemke | Bimatrix equilibrium points and mathematical programming[END_REF] parametric algorithm used by mathematicians to solve linear complementarity problems.

A return to Ricardo's original approach does enrich the results drawn from a static analysis: if, for a given level of demand, multiple equilibria are found, the ultimate reason is that the dynamics fail somewhere on a path which leads from low levels of demand to the present level. Similarly, an unexpected distribution of income for a given demand is due the presence of non-Ricardian equilibria on that trajectory.

CONCLUSION

Ricardo's message can be interpreted in a weak sense: the price of a resource goes to infinity if it becomes scarcer and scarcer and admits no substitute, with significant effects on distribution. The Ricardian dynamics are more precise and ambitious, and Ricardo's views on the problem explain the structure of the Principles: Chapter I deals with the value of reproducible commodities and allows him to establish the basic trade-off property between wages and profits in the absence of lands. Chapter II explains why rents are inessential (only from an analytical point of view!) and can be ignored by considering the conditions of production in industry and the marginal conditions in agriculture. But Ricardo did not draw the consequences of an exclusive determination of the incoming marginal method by a profitability criterion. When they are taken into account, the smooth path of the dynamics may become rather chaotic, with either some steps during which the expected laws of distribution do not apply or