Hippolyte D'albis

Johanna Etner
email: johanna.etner@u-paris10.fr

Jean-Pierre Drugeon

Ben Heijdra

Jochen Mierau

Illiquid Life Annuities 1

In this article, we consider illiquid life annuity contracts and show that they may be preferred to Yaari (1965)'s liquid contracts. In an overlapping-generation economy, liquid life annuities are demanded only if the equilibrium is dynamically inefficient. Alternatively, an equilibrium displaying a positive demand for illiquid life annuities is efficient. In this latter case, the welfare at steady-state is larger if illiquid life annuity contracts are available.

Introduction

In this article, we challenge the common thought that the life annuity contract proposed by Yaari in his seminal 1965's paper is optimal. We indeed show, in a standard neo-classical framework, that another contract, which actually resembles much more to the contracts offered by insurance companies, may be preferred by rational individuals.

The economic theory of annuities has been strongly influenced by [START_REF] Yaari | Uncertain lifetime, life insurance, and the theory of the consumer[END_REF]. He studies the optimal demand for annuities in a life-cycle model with or without bequest motives. The financial asset that is named annuity by Yaari has the following characteristics: the returns are positive if the bearer is alive and zero if he is not. Annuities are nevertheless demanded since their returns are larger than the one yielded by risk-free bonds. The difference between the two yields is the annuity premium, which is said to be fair when it equals the inverse of the survival probability. Importantly, as the individual ages, the premium increases. This characterization of an annuity has been quite influential and has lead to numerous studies (See among others [START_REF] Davidoff | Annuities and individual welfare[END_REF][START_REF] Sheshinski | The Economic Theory of Annuities[END_REF].

Many types of annuity contracts exist [START_REF] Cannon | Annuity Markets[END_REF]. Their common features are quite different from Yaari's annuities. First, the premium is age-independent. The individual purchases some annuities during youth and, at a given age -let say the age at retirement-he periodically receives a fixed amount as long as he survives. Second, the contract is irreversible. Once payments have begun, one can not recover the amount invested. An implicit assumption in Yaari is that agents, upon survival, receive the capital and the interests of their annuity. This means that they are in position to renegotiate their contract at each period, and that is why the premium increases as the individual ages.

In this article, we propose a standard framework in which the individual has the choice between two types of life annuity contract. The first one that we named flexible, is the one proposed by [START_REF] Yaari | Uncertain lifetime, life insurance, and the theory of the consumer[END_REF]. The second one, which is named illiquid, is irreversible and proposes age-independent returns. In both case, we suppose that the annuity premium is such that the insurance companies make no profit. Illiquid annuities have been introduced in life-cycle models by [START_REF] Horneff | Life-cycle asset allocation with annuity markets[END_REF] and [START_REF] Peijnenburg | The annuity puzzle remains a puzzle[END_REF] in order to discuss the issue 1 about the low demand for annuities. Our purpose is to study analytically the equilibrium and welfare consequences of the existence of such contracts.

First, we analyze the life-cycle optimal decision under uncertain lifetime.

Importantly, we consider a setting in which the individual ages, which more precisely means that survival probabilities decrease with age. We therefore depart from two-period life-cycle models or from [START_REF] Blanchard | Debt, deficits and finite horizons[END_REF]'s setting in which our distinction between increasing and fixed returns makes no sense. We obtain that illiquid annuity are preferred to flexible one if the expected returns of the first are sufficiently greater than those of the second. This is the consequence of an arbitrage between more flexibility and more returns.

Second, we consider the general equilibrium of our economy, in which returns of both contracts are determined by the markets. We study a simple overlapping generation economy, similar to the one analyzed by [START_REF] Diamond | National debt in a neoclassical growth model[END_REF]. Surprisingly, we show that illiquid annuities are preferred when the equilibrium is dynamically efficient while flexible annuities are preferred when it is inefficient.

We then discuss about the optimality of both annuity contracts. In particular, for dynamically efficient equilibrium, the welfare at steady-state is larger if illiquid life annuity contracts are available.

Finally, to test the robustness of our results, we propose three extensions of our model by considering successively a background risk, a bequest motive and a subjective evaluation of survival probabilities.

Individual behavior 2.1 Demographics

We consider an overlapping generations model in which agents live a finite and uncertain length of time. They live for a maximum of three periods, also called ages, which are denoted i = {0, 1, 2}. The probability of being alive at age i, conditional on survival until age i -1, is denoted p i . Survival probabilities at each age are constant over time, but decrease with age.

Let N i,t be the number of agents of age i at time t. At each time t = 0, 1, 2..., N 0,t identical agents are born. Thus, the number of agents of age 1 born at time t is N 1,t+1 = p 1 N 0,t and the number of agents of age 2 born at time t is N 2,t+2 = p 1 p 2 N 0,t . Finally, we assume that the number of agents of age 0 grows at a constant growth rate, denoted n, with n > -1:

N 0,t = (1 + n) N 0,t-1 .
(1)

Annuity markets

Agents can invest in two types of financial products: bonds and life annuities.

The yield on bonds is risk free: each unit of consumption invested at time t -1 yields R t units of consumption at time t. Concerning annuities, two types of contracts are offered by insurance companies. It is assumed that information on the probability of survival is perfect and that markets for each contract are competitive, which implies that the proposed contracts are fair. It is further assumed that a company cannot cross-subsidize the types of contracts it offers.

All these assumptions imply that the profit of insurance companies is zero for each contract. We will now explain in detail the characteristics of both annuity contracts.

The first annuity contract offered to agents is that found in most articles of the literature since the seminal article of [START_REF] Yaari | Uncertain lifetime, life insurance, and the theory of the consumer[END_REF]. This is an actuarially fair contract that can be renegotiated each time. We denote a 0,t and a 1,t+1 as the demands for flexible annuities at ages 0 and 1 by an agent born at time t. At age 2, the demand for annuities must be zero because the agent has reached, by assumption, the last period of life.

The second annuity contract proposed to agents has the following features: the investment must be made at age 0, the capital cannot be recovered before age 2 and the remuneration received is independent of age. Annuity is said to be illiquid because, at age 1, the agent receives only the interest of his investment.

Equivalently, it can be said that the agent must invest at age 1 the same amount that he invested at age 0. We denote b t as the demand for illiquid annuities by an agent of age 0 at time t. To calculate the annuity yield, the condition of zero profit for insurance companies is applied. The companies collect at time t -1 the agent's savings and invest them at the risk-free rate. At time t, the value of this investment, which is equal to

(N 0,t-1 b t-1 + N 1,t-1 b t-2 ) R t , (2)
is redistributed among the surviving agents. If we denote R t /π t as the yield at time t for each unit of consumption invested in t -1 or t -2, we conclude that the amount distributed must be equal to

(N 1,t b t-1 + N 2,t b t-2 ) R t π t . (3)
Consequently, by equalizing ( 2) and ( 3), the inverse of the premium solves:

π t = p 1 (1 + n) b t-1 + p 1 p 2 b t-2 (1 + n) b t-1 + p 1 b t-2 . (4)
If the demands for illiquid annuities are positive, it is easy to show that π t ∈ [p 2 , p 1 ]. We conclude that the interest paid at age 1 is higher than that of the flexible annuities, while the interest paid at age 2 is lower. Hence, the flexible contract is more profitable the older the agent and illiquid annuities can be interpreted as an intergenerational transfer from agents age 2 to agents age 1.

This explains why the yield R t /π t is a decreasing function of the population growth rate, n. Finally, we note that in the limit case p 1 = p 2 , the yields of the two annuity contracts are equal.

Life-cycle choices

Each agent chooses a portfolio and a savings strategy to achieve an optimal consumption allocation between the different ages. The intertemporal expected utility of an agent of age 0 at time t reads as:

u (c 0,t ) + θp 1 u (c 1,t+1 ) + θ 2 p 1 p 2 u (c 2,t+2 ) , (5)
where c i,t+i is the consumption at age i, and θ > 0 is a discount factor. The instantaneous utility function u, is increasing and concave, u > 0 and u < 0, and is such that lim x→0 u (x) = +∞ and lim x→+∞ u (x) = 0. Budget constraints are as follows: at time t, the agent of age 0 receives a wage, denoted w t , which he allocates between consumption and savings. It may consist of flexible annuities, a 0,t , and illiquid annuity, b t . The budget constraint at age 0 is:

c 0,t = w t -a 0,t -b t . (6)
We notice that investment in risk-free bonds is not modelled here because it is never an optimal strategy. Furthermore, short selling constraints are imposed on both investments, which together with positivity constraints on consumption allow us to eliminate degenerate strategies.

a 0,t ≥ 0, b t ≥ 0, c 0,t ≥ 0, c 1,t+1 ≥ 0, c 2,t+2 ≥ 0. ( 7
)
At time t + 1, the agent receives the capital and interest of his flexible annuity investment and the interest of his illiquid annuity investment. These financial revenues are used by the agent to finance his consumption and savings in the form of flexible annuities, for which the demand is denoted a 1,t+1 . The budget constraint at age 1 is:

c 1,t+1 = a 0,t R t+1 p 1 + b t R t+1 π t+1 -1 -a 1,t+1 . (8)
At time t + 2, which corresponds to the last period of life of the agent, consumption is equal to the capital and interest of his flexible and illiquid annuity investments. The bounded lifespan hypothesis implies that the capital invested in illiquid life annuity is recovered at age 21 . The budget constraint at age 2 is:

c 2,t+2 = a 1,t+1 R t+2 p 2 + b t R t+2 π t+2 . (9)
The problem of the agent is to choose {c 0,t , c 1,t+1 , c 2,t+2 , a 0,t , b t , a 1,t+1 } that maximizes (5) subject to ( 6), ( 7), ( 8) and ( 9). Let us denote:

R t+1 := p 1 π t+1 R t+1 -p 1 1 - p 2 π t+2 . ( 10
)
Our first result is the following.

Proposition 1. The optimal portfolio satisfies:

b t > 0 and a 0,t = 0 if R t+1 > R t+1 b t = 0 and a 0,t > 0 if R t+1 < R t+1 (11)
A portfolio satisfying b t > 0 and a 0,t > 0 can be optimal only if R t+1 = R t+1 .

Proof. See Appendix.

In this first version of the model, without background risks nor borrowing constraints, illiquidity is not an issue specific to the agent. Portfolio choice is therefore based on a comparison of the respective yields from flexible annuities and illiquid annuities. Relevant yields are expected yields, discounted at the risk-free interest rate and calculated assuming no reinvestment of the interest received at age 1 2 .

Through π t+1 and π t+2 given in (4), we see that Rt+1 is affected by the demands for annuities by past and future generations. In particular, R t+1 increases with b t-1 and decreases with b t+1 . Because the illiquid annuity contract acts as a transfer from the oldest to the youngest, the more it is demanded by the previous generation, the more the comparative advantage increases, but the more it is demanded by the next generation, the more the comparative advantage decreases. We also note that the yields of the two contracts are equal in the limit case p 1 = p 2 .

With Proposition 1, we have seen that the portfolio is generically composed of a single type of contract. The optimal consumption allocation of the agent then depends on the chosen contract. If flexible annuities are chosen at age 0, the result is typical of that found in the literature: consumption dynamics are independent of survival probabilities and increase according to the ratio of the interest factor over the discount factor [START_REF] Yaari | Uncertain lifetime, life insurance, and the theory of the consumer[END_REF]. Conversely, if illiquid annuities are chosen, the optimal consumption dynamics can be characterized by the following proposition.

Proposition 2. Suppose that R t+1 > R t+1 . The optimal consumption allocation satisfies:

u (c 0,t ) u (c 1,t+1 ) > u (c 1,t+1 ) u (c 2,t+2 ) if R t+1 ≥ R t+2 . ( 12
)
Thus, if the utility function is homogenous, inequality (12) can be rewritten as: 2 The condition R t+1 ≥ Rt+1 can indeed be rewritten as:

p 1 R t+1 R t+1 p 1 + p 1 p 2 R t+1 R t+2 R t+2 p 2 ≥ p 1 R t+1 R t+1 π t+1 + p 1 p 2 R t+1 R t+2 R t+2 π t+2 . c 1,t+1 c 0,t > c 2,t+2 c 1,t+1 if R t+1 ≥ R t+2 . (13)
Proof. See Appendix.

Provided that the interest rate is not increasing and that the utility function has standard properties, the holding of a portfolio composed of illiquid annuities implies that the consumption growth rate decreases with age. This is explained by the fact that the marginal rate of substitution (MRS) between ages 0 and 1 is given by R t+1 , which is higher than R t+1 (as shown in Proposition 1), turns to be greater than the MRS between ages 1 and 2, which is given by R t+2 .

Between and 1 is affected by the survival probabilities even if the intertemporal utility function is additively separable 4 . For a given and constant demand for annuities, it can be shown that the relationship is positive if the interest rate is higher than the population growth rate.

In this section, we have shown that there exists a set of interest rate values for which illiquid annuities are purchased by agents. In the next section, we analyze the choice of agents when prices are determined by the equilibrium conditions in all markets.

3 General equilibrium analysis

Annuities and the efficiency of the equilibrium

The production side of the model is standard. There exists a unique good that is produced by many firms acting on a perfectly competitive market. The produc-tion function displays constant returns-to-scale and satisfies Inada conditions.

We assume that only agents of age 0 are working and denote by k t the capital stock per worker at time t. Assuming that capital depreciation rate is 100% per period, the optimality conditions of the firms can be written as:

w t = f (k t ) -k t f (k t ) and R t = f (k t ) . ( 14
)
The equilibrium condition on the capital market is satisfied if the capital stock at time t + 1 is equal to the sum of the savings of agents born at times t and t -1. This condition can be written as:

k t+1 = a 0,t + b t 1 + n + p 1 (a 1,t + b t-1 ) (1 + n) 2 . ( 15
)
In what follows, we assume there exists a unique steady-state. Depending on the model parameters values, the interest rate at steady-state may be higher or lower than the population growth rate. It is well known5 that a converging trajectory to such steady states is efficient in the first case and inefficient otherwise.

Let a stared variable denote the steady-state equilibrium value of the considered variable. The following proposition, which is the counterpart at equilibrium of Proposition 1, characterizes the portfolio choices of agents based on the efficiency of the steady state.

Proposition 3. At steady-state, the optimal portfolio satisfies:

   b * > 0 and a * 0 = 0 if f (k * ) > 1 + n, b * = 0and a * 0 > 0 if f (k * ) < 1 + n. (16)
A portfolio satisfying b * > 0 and a * 0 > 0 can be optimal only if f (k * ) = 1 + n. Proof. See Appendix. Proposition 3 states that if the equilibrium is dynamically efficient, the agents hold illiquid annuities in the steady state. It is only when the equilibrium is inefficient that they are not held. The proof is simple and is based on the difference between the yields offered by flexible and illiquid annuities.

Using equations ( 4), ( 10) and ( 14), written in the steady state, we observe that R * can be written as a linear function of the marginal productivity of capital:

R * = 1 + n + p 1 1 + n + p 2 f (k * ) - (p 1 -p 2 ) (1 + n + p 2 ) 1 + n + p 2 (17)
The yield on illiquid annuities is greater than on flexible life annuity if and only if it is greater than 1 + n, the population growth factor. The figure below shows the spread in yields as a function of the steady state interest factor R * .

Insert Figure 1.

The intuition behind the result stated in Proposition 3 is based on the fact that illiquid annuity represents a transfer from one generation to the next generation. When the population growth rate is relatively low, which is the case when the equilibrium is efficient, this transfer is inexpensive and the investment is profitable. Conversely, when the growth rate is high, illiquid annuity investment is unprofitable. Finally, at the Golden Rule, flexible and illiquid annuities have exactly the same profitability. Somehow, illiquid life annuity is the opposite of a Pay-As-You-Go pension system, which is a transfer to the previous generation and a profitable investment when the equilibrium is inefficient.

Annuities and the welfare at steady-state

The next step concerns the welfare of an agent in the steady state. We have seen that when the equilibrium is efficient, illiquid annuity is preferred to flexible annuity. This has been established for an equilibrium interest rate associated to the level of capital per worker at equilibrium. It does not, however, take into account the fact that the capital per worker may be different in an economy where illiquid annuities are proposed and in an economy where they are not.

So to evaluate the effect of the supply of illiquid annuity contracts on welfare, we proceed as follows: we compare the welfare obtained in an economy where the two types of contracts are offered to welfare obtained in an economy where only flexible annuity is available. The result of this comparison is presented in the following proposition. Proof. See Appendix.

In the proof of Proposition 4, we show that the introduction of illiquid annuity contracts increases the capital per worker in the steady state. The intuition for this result is the following: as it induces a shift to youth, illiquid annuity stimulates savings. This increase is conducive to steady-state welfare when the equilibrium is inefficient, as the utility increases with capital in that case. The proof of Proposition 4 is based on the assumption of the existence of a unique steady state. In the case of multiple equilibria, the same comparison can be made using the stability properties.

In the long run, agents benefit from the existence of an illiquid annuity market provided that the equilibrium is efficient. However, the existence of an illiquid annuity market in the steady state depends on the decisions made by agents along the transitory path. This is demonstrated in the following proposition.

Proposition 5. Illiquid annuity contracts are offered in the steady state only if all previous generations have purchased illiquid annuity.

Proof. See Appendix.

Proposition 5 shows that the Pareto optimality of illiquid annuity contracts at steady-state is not a sufficient condition for the existence of such a market.

While the generation born in t chooses not to invest in illiquid annuity, we see, by using equation ( 4), that the preceding generation benefits at age 2 from a yield equal to R t+1 /p 2 , equal to the one of flexible annuity, and the generation that follows should settle at age 1 for a yield equal to R t+2 /p 1 . As this yield is equal to one of flexible annuity, the generation born at t + 1 has no interest in investing in illiquid annuity, after which the contract is never requested.

We conclude that if it exists, the illiquid annuity contract represents a Pareto improvement for all generations.

Equivalently, we can notice that illiquid annuities will never be demanded if the contract has not been proposed before the initial time of the economy t = 0, that is to say, if agents born at t = -1 do not have illiquid annuities in their portfolio at t = 0. If the contract does not initially exist, it will not appear spontaneously in a market economy. This fact makes it necessary to intervene in order to possibly compensate for earlier generations to increase the welfare of future generations. Although it is not sufficient, this result may also help understanding the low participation in annuity markets (for a recent survey on the annuity puzzle, see [START_REF] Benartzi | Annuitization Puzzles[END_REF].

In this section, we have presented the conditions for the existence of an illiquid annuity market and demonstrated the Pareto improvement that it generates.

In the next section, we discuss the robustness of our results.

Robustness

The results presented above are not changed if we consider alternative assumptions about the agents' preferences and the environment in which they make their decisions. We consider, in particular, a non-borrowing constraint at age 1, possibly with a background risk that may affect consumption at ages 1 and 2, an assumption of bequest motivated by joy-of-giving, and finally, a subjective evaluation of the survival probabilities. We show that in all these cases, Proposition 1 is not, or barely, changed.

The first extension we consider is a non-borrowing constraint at age 1. In our framework, this implies that selling annuities short, or equivalently purshasing life insurance contracts [START_REF] Bernheim | How strong are bequest motives? Evidence based on estimates of the demand for live insurance and annuities[END_REF], is not allowed. We therefore add the following inequality to the optimization problem described above:

a 1,t+1 ≥ 0. (18)
The Proposition 1 is modified as follows.

Proposition 6. Let the agent maximizes (5) subject to ( 6), ( 7), ( 8), ( 9) and [START_REF] Gourinchas | Consumption over the life cycle[END_REF]. The optimal portfolio satisfies:

   b t > 0 if R t+1 > R t+1 , b t = 0, a 0,t > 0 and a 1,t > 0 if R t+1 < R t+1 . (19)
A portfolio satisfying b t > 0 and a 0,t > 0 can be optimal only if R t+1 ≥ R t+1 .

In the case R t+1 > R t+1 , a 0,t > 0 can be optimal only if a 1,t+1 = 0.

Proof. See Appendix.

With proposition 6, we see that introducing a non-borrowing constraint at age 1 barely modifies the optimal portfolio. Constraint ( 18) is binding only if the demand for illiquid annuities is positive, as the consumption at age 2 would be otherwise zero. Provided that constraint ( 18) is binding, the MRS between ages 0 and 1 is still greater than R t+1 while remaining lower than R t+1 , whereas between ages 1 and 2 is greater than R t+2 . In a nutshell, it is the dynamics of consumption that is modified by the non-borrowing constraint, not the optimal portfolio.

Let us now introduce a background risk that may reduce consumptions at ages 1 and 2. This risk can be interpreted as health shocks that require costly treatments and against with it is not possible to be insured 6 . Together with the constraint [START_REF] Gourinchas | Consumption over the life cycle[END_REF], this shock makes the annuity contract non flexible [START_REF] Direr | Flexible life annuities[END_REF]). Consumptions at ages 1 and 2 are then written as random variables, denoted c1,t+1 and c2,t+2 , and the expected utility of the agent of age 0 at time t reads as:

u (c 0,t ) + θp 1 Eu (c 1,t+1 ) + θ 2 p 1 p 2 Eu (c 2,t+2 ) . (20)
The optimal behavior of the agent is given in the following.

Proposition 7. Let the agent maximizes [START_REF] Horneff | Life-cycle asset allocation with annuity markets[END_REF] subject to ( 6), ( 7), ( 8), ( 9) and [START_REF] Gourinchas | Consumption over the life cycle[END_REF]. The optimal portfolio satisfies the same conditions as those described in Proposition 6.

Proof. See Appendix.

As the portfolio choice depends on a comparison of yields, it is not affected by considering random utilities.

The second extension we consider is a bequest motive. The investment in regular bonds can indeed be justified on the grounds of intergenerational altruism and, as shown by [START_REF] Lockwood | Bequest motives and the annuity puzzle[END_REF], this may help explaining the low demand for annuities. Following [START_REF] Yaari | Uncertain lifetime, life insurance, and the theory of the consumer[END_REF], the bonds held in the portfolio at the age of death are bequested, and the utility of the agent increases with the amount that is bequested. As in [START_REF] Davidoff | Annuities and individual welfare[END_REF], we suppose that capitalized value of the bequest enter the expected utility, which reads as:

u (c 0,t ) + θp 1 u (c 1,t+1 ) + θ 2 p 1 p 2 u (c 2,t+2 ) + (1 -p 1 ) v (R t+1 R t+2 h 0,t ) +p 1 (1 -p 2 ) v (R t+2 h 1,t+1 ) + p 1 p 2 v (h 2,t+2 ) , (21)
where h i,t+i is the demand for bonds made by an agent of age i, i = {0, 1, 2}, as of time t + i. Function v is increasing and concave and we assume that lim x→0 v (x) = +∞, which restrict our analysis to interior solutions. Bonds' yield is the risk-free rate. Thus, the budget constraints ( 6), ( 8), ( 9) are replaced by the following ones:

c 0,t = w t -a 0,t -b t -h 0,t , (22)
c 1,t+1 = a 0,t R t+1 p 1 + b t R t+1 π t+1 -1 + h 0,t R t+1 -a 1,t+1 -h 1,t+1 , (23)
c 2,t+2 = a 1,t+1 R t+2 p 2 + b t R t+2 π t+2 + h 1,t+1 R t+2 -h 2,t+2 . (24)
The optimal behavior of the agent is given in the following.

Proposition 8. Let the agent maximizes [START_REF] Lockwood | Bequest motives and the annuity puzzle[END_REF] subject to ( 7), ( 22), [START_REF] Savage | The Foundations of Statistics[END_REF], and [START_REF] Sheshinski | The Economic Theory of Annuities[END_REF]. The optimal portfolio satisfies conditions [START_REF] Davidoff | Annuities and individual welfare[END_REF]. Moreover, the capitalized bequests are such that:

   R t+1 R t+2 h 0,t = R t+2 h 1,t+1 = h 2,t+2 if b t = 0, R t+1 R t+2 h 0,t < R t+2 h 1,t+1 = h 2,t+2 if a 0,t = 0. (25)
Proof. See Appendix.

The introduction of a joy-of-giving altruistic motive modifies the optimal porfolio as regular bonds are demanded in order to be bequested. However, the remaining of the optimal portfolio is composed of flexible annuities for R t+1 < R t+1 and of illiquid annuities for R t+1 > R t+1 . With flexible annuities, the optimal tradeoff between consumption and bequest is the same as in The third extension considers a subjective evaluation of the survival probabilities. Many studies have indeed demonstrated the importance of probability distortion in risky choices, and notably when the risk at stake concerns health and longevity [START_REF] Brewer | Meta-analysis of the relationship between risk perception and health behavior: The example of vaccination[END_REF]. We consider an agent endowed with subjective survival probabilities, denoted p1 and p2 , which are such that p1 = p 1 and p2 = p 2 . His preferences are represented by the following subjective expected utility 7 :

u (c 0,t ) + p1 θu (c 1,t+1 ) + p2 θ 2 u (c 2,t+2 ) . ( 26
)
The rest of the model is the same as in section 2.3, which implies that agent's beliefs differ from the insurers' survival probabilities estimation.To simplify, we therefore do not take into account the possibility for insurers to use this information and modify annuity's yields. The optimal behavior of the agent is given in the following.

Proposition 9. Let the agent maximizes ( 26) subject to ( 6), ( 7), [START_REF] Cannon | Annuity Markets[END_REF], and ( 9).

The optimal portfolio satisfies conditions [START_REF] Davidoff | Annuities and individual welfare[END_REF].

Proof. See Appendix.

Once again, our main results are robust. Introducing a subjective evaluation of longevity risk does not modifies the preference for illiquid annuity as long as their objective yield is sufficiently large.

Conclusion

In this paper, we showed that illiquid annuity is preferred to flexible one provided that the equilibrium is dynamically efficient. Moreover, the availability of illiquid annuity permits a welfare improvement in the long run. Nevertheless, they are offered in the steady-state only if all generations have purchased them in the past. Consequently, policy intervention can be justified even if the equilibrium is efficient.

This study can be extended in several directions. First a multi-period setting can be analyzed in order to investigate the issue of the optimal timing of annuity purchase [START_REF] Brugiavini | Uncertainty resolution and the timing of annuity purchases[END_REF] and discuss the opportunity represented by deferred annuities. Second, heterogeneous agents could be introduced in order to focus on adverse selection (Bommier et al., 2011) and redistribution issues [START_REF] Cremer | Collective annuities and redistribution[END_REF]. Finally, aggregate risk on mortality (Schulze and Post, 2010) as well as other aggregate risk could be introduced in order to discuss the risk sharing properties [START_REF] Gollier | Intergenerational risk-sharing and risk-taking of a pension fund[END_REF] of the illiquid annuities we considered.

where:

       c 1 = [f(k)-kf (k)-c0] p1 f (k)(1+n+p1) 1+n+p2 + (1 + n) -(1+n) 2 k p1 , c 2 = (1+n) 2 p1p2 kf (k) - [f(k)-kf (k)-c0]f (k)(1+n+p1)(1+n) p1p2(1+n+p2) . (42)
Let us consider now an economy where only flexible annuities are proposed. The steady-state of such an economy solves:

                                     c 0 = f (k) -kf (k) -a 0 , c 1 = a 0 f (k) p1 -a 1 , c 2 = a 1 f (k) p2 , k = a0 1+n + p1a1 (1+n) 2 , 0 = u (c 0 ) -θf (k) u (c 1 ) , 0 = u (c 1 ) -θf (k) u (c 2 ) . (43)
System (43) reduces to a system in (c 0 , k) that reads as:

   -u (c 0 ) + θf (k) u (c 1 ) = 0, -u (c 1 ) + θf (k) u (c 2 ) = 0, (44)
where:

     c 1 = [f (k) -kf (k) -c 0 ] f (k)+(1+n) p1 -(1+n) 2 k p1 , c 2 = {(1 + n) k -[f (k) -kf (k) -c 0 ]} (1+n)f (k) p1p2 . ( 45
)
The objective is thus to compare the steady-state capital that is the solution of (41) with the one that is solution of (44). To do so, we set up, for z ∈ [1, (1 + n + p 1 ) / (1 + n + p 2 )], a more general system that writes:

   -u (f (k) -kf (k) -x) + θ [f (k) z -(1 + n) (z -1)] u (c 1 ) = 0, -u (c 1 ) + θf (k) u (c 2 ) = 0, (46)
where

             x = f (k) -kf (k) -c 0 , c 1 = x[f (k)z+(1+n)] p1 -(1+n) 2 k p1 , c 2 = [(1 + n) k -xz] (1+n)f (k) p1p2 . (47)
We notice that for z = 1, system (46) reduces to system (44) while for z =

(1 + n + p 1 ) / (1 + n + p 2 ), system (46) reduces to system (41). To prove our claim, we hence aim at showing that:

dk * dz > 0, ( 48
)
where k * is the capital stock that is the solution of (46). Let us rewrite the first equation in (46) as F (x, k; z) = 0 and the second as G (x, k; z) = 0. One has:

dk dz = - F z - G z G x F x F k - G k G x F x . ( 49
)
Consider first the numerator of (49). Let σ (c) := -u (c) /cu (c). Simple computations give that the sign of F z -G z F x /G x is the same as the one of:

f (k) -(1 + n) f (k) z -(1 + n) (z -1) × 1 σ (c 1 ) f (k) z + (1 + n) x [f (k) z + (1 + n)] -(1 + n) 2 k + 1 σ (c 2 ) z (1 + n) k -xz + 1 c 0 σ (c 0 ) 1 σ (c 1 ) xf (k) x [f (k) z + (1 + n)] -(1 + n) 2 k + 1 c 0 σ (c 0 ) 1 σ (c 2 ) x (1 + n) k -xz + 1 σ (c 1 ) 1 σ (c 2 ) x (1 + n) k -xz (1 + n) x [f (k) z + (1 + n)] -(1 + n) 2 k , (50)
which is positive as we supposed that f (k) > (1 + n). To determine the sign of the denominator, we use the assumption of the existence of a unique equilibrium.

System (46) can be written as a single dimension problem: F (φ (k; z) , k; z) = 0 where φ (.) is the implicit function obtained using G (x, k; z) = 0. The derivative of F (φ (k; z) , k; z) with respect to k is given by F k -G k F x /G x . As F (φ (0; z) , 0; z) > 0, we conclude that the derivative, computed at the equilibrium k * is negative. Using (49), we finally conclude that dk/dz > 0.

Step 2. We now compute the derivative of the intertemporal utility function with respect to capital, such as

u (c 0 ) dc 0 dk + θp 1 u (c 1 ) dc 1 dk + θ 2 p 1 p 2 u (c 2 ) dc 2 dk , (51)
In steady-state of an economy where both flexible and illiquid annuities are proposed, we use (41) and (42) to obtain that the sign of (51) is the same as the one of:

[f (k) -(1 + n)] k (1 + n) -[f (k) -2kf (k) -c 0 ] 1 + n + p 1 1 + n + p 2 . (52)

  ages 1 and 2, all additional savings are indeed invested in flexible annuities. The lower yield of investment opportunities when the agent ages can explain the decrease in the growth of consumption. Introducing illiquid annuities in a life-cycle model allows better reproduction of the stylized facts of the individual's consumption during his life cycle (see, e.g., Gourinchas and Parker, 2002 and Fernández-Villaverde and Krueger, 2007) even though annuities are fairly priced 3 . It should be noted, moreover, that the MRS between periods 0

Proposition 4 .

 4 Let f (k * ) > 1 + n. The welfare at steady-state is larger if illiquid annuity contracts are available.

Figure 1 . 6 -

 16 Figure 1. Spread in yields at steady state

This condition is the counterpart of a transversality condition that should be introduced in a more realistic model with a large number of periods of life where survival probabilities converge to 0 when age tends to infinity.

Alternatively, a concave consumption can be obtained if annuities are not available[START_REF] Davis | Uncertain lifetime, consumption, and dissaving in retirement[END_REF] or not fairly priced[START_REF] Hansen | Consumption over the life cycle: The role of annuities[END_REF].

[START_REF] Bommier | Uncertain lifetime and intertemporal choice: Risk aversion as a rationale for time discounting[END_REF] obtained the same kind of result with flexible annuities under the condition of non separability of the utility function.

Cass 1972, De La Croix and Michel 2002, page 83

Long-Term care can be though as an example of this even though insurance contracts are offered in some countries. See[START_REF] Brown | Insuring long-term care in the United States[END_REF].

Although different models of representation of preferences under uncertainty have been proposed, in case of two states of nature, the main models reach to one, namely the subjective model[START_REF] Savage | The Foundations of Statistics[END_REF].

for insightful comments. Financial support from the European Research Council (ERC Stg Grant 283953

Appendix

Proof of Proposition 1. We denote µ t as the Kuhn-Tucker multiplier associated with the non-negativity constraint: a 0,t ≥ 0, and λ t as the one associated with: b t ≥ 0. The first order conditions of the optimization problem can be written as:

while the complementary slackness conditions are: µ t a 0,t = 0 and λ t b t = 0. (28)

By rearranging equations in system (27), we obtain:

where R t+1 is defined in [START_REF] Cremer | Collective annuities and redistribution[END_REF].

Let us first notice that having both λ t > 0 and µ t > 0 is not possible as we can see, using the complementary slackness conditions (28) and the budget constraints ( 8) and [START_REF] Cass | On capital accumulation in the aggregate, neoclassical model of economic growth: a complete characterization[END_REF], that this would imply:

which contradicts the fact that optimal consumptions should be positive. As a consequence, we use (29) to state that:

which, using the complementary slackness conditions (28), allow us to conclude the proof.

Proof of Proposition 2. For R t+1 > R t+1 , we have seen in the proof of Proposition 1 that λ t = 0 and µ t > 0. Thus, the last two equations of system (27) can be rewritten as follows:

Thus, we have:

which, using that fact that R t+1 > R t+1 , allow us to write [START_REF] Davis | Uncertain lifetime, consumption, and dissaving in retirement[END_REF]. To obtain [START_REF] De La Croix | A Theory of Economic Growth[END_REF],

we us the fact that if u is homogenous of degree κ + 1, u is homogenous of degree κ, which implies that (33) can be rewritten as follows:

Proof of Proposition 3. An intertemporal equilibrium is a collection:

which satisfies the budget constraints ( 6), ( 8) and ( 9), the optimality conditions ( 14) and ( 27), the complementary slackness conditions (28), the zero-profit condition (4) and the equilibrium condition [START_REF] Direr | Flexible life annuities[END_REF]. At steady state, the equilibrium is the solution of the following system:

as well as ( 4) and [START_REF] Diamond | National debt in a neoclassical growth model[END_REF].

From Proposition 1, we know that the possible portfolio at steady-state are:

(1) a * 0 > 0 and b * > 0, (2) a * 0 = 0 and b * > 0, (3) a * 0 > 0 and b * > 0. Let us consider those three cases successively.

For a * 0 > 0 and b * > 0, the last three equations of (36) can be rewritten as:

which is satisfied for f (k) = 1 + n, i.e. when the capital is at the Golden Rule.

For a * 0 = 0 and b * > 0, the last three equations of (36) can be rewritten as:

which are satisfied only if f (k) ≥ 1+n. Using what has been shown just above, we conclude that if f (k) > 1 + n, one has a * 0 = 0. Finally, for a * 0 > 0 and b * = 0, the last three equations of (36) can be rewritten as:

Proof of Proposition 4. The proof proceeds in two steps. In step 1, we show that the capital stock is higher in an economy where flexible and illiquid annuities are proposed than in an economy where only flexible annuities are proposed. In step 2, we show that the utility increases with the capital stock.

Step 1. Let us consider first an economy where flexible and illiquid annuities are proposed. If f (k) > 1 + n, we can use the proof of Proposition 3 to state that the steady-state solves:

System (40) reduces to a system in (c 0 , k) that reads as:

Using the fact that c 2 , whose expression is given in (42), is positive we conclude that (52) is positive. Hence, an increase in capital increases the welfare of the agent in the steady-state.

Proof of Proposition 5. The objective is to prove that if there exists T such that b T -1 = 0 then b T +i = 0 for all i = 0, 1, 2, ... To prove it, we consider the yield of the investment in illiquid annuities made at time T . Replacing ( 4) and b T -1 = 0 in [START_REF] Cremer | Collective annuities and redistribution[END_REF], we obtain:

For b T +1 > 0, we obtain that R T +1 > R T +1 , which implies, using Proposition 1, that b T = 0.

Proof of Proposition 6. The proof is similar to the one of Proposition 1. We denote (µ t , λ t , γ t ) as the Kuhn-Tucker multipliers associated with the nonnegativity constraints: a 0,t ≥ 0, b t ≥ 0 and a 1,t ≥ 0. The first order conditions of the optimization problem can be written as:

while the complementary slackness conditions are:

By rearranging equations in system (54), we obtain:

where R t+1 is defined in [START_REF] Cremer | Collective annuities and redistribution[END_REF].

Let us consider the various configurations that are possible. As in the proof of Proposition 1, the case λ t > 0 and µ t > 0 is not optimal as it implies that the sign of c 1,t+1 is the opposite of the one of c 2,t+2 . Similarly, the case λ t > 0 and γ t+1 > 0 is neither optimal as it implies c 2,t+2 = 0. We now use equation (56) to establish that:

-for R t+1 > R t+1 , one has λ t < µ t + γ t+1 p 2 /π t+2 , which necessarily implies:

However, b t = 0 is not possible as the positivity of c 1,t+1 would thus imply a 0,t > 0 (and µ t = 0) while the positivity of c 2,t+2 would imply a 1,t+1 > 0 (and γ t+1 = 0). Thus, b t > 0. Moreover, µ t ≥ 0 and γ t+1 ≥ 0, with at least one of the two inequalities being strict.

-for R t+1 = R t+1 , one has λ t = µ t + γ t+1 p 2 /π t+2 , which necessarily implies:

-for R t+1 > R t+1 , one has λ t > µ t + γ t+1 p 2 /π t+2 , which necessarily implies:

λ t > 0 and µ t = γ t+1 = 0. Due to (55) we conclude that b t = 0 while the positivity of c 1,t+1 implies a 0,t > 0 and the positivity of c 2,t+2 implies a 1,t+1 > 0.

Proof of Proposition 7. Following the same derivations as those made in the proof of Proposition 6, we obtain:

which is the counterpart of (56). The reasonning made after (56) also applies here.

Proof of Proposition 8. The first order conditions of the agent's problem are given by ( 27), (28) and:

As a consequence (29) and (31) still hold. Moreover, by replacing the first and the third equations of ( 27) in (58), we obtain:

This allow us to conclude that:

Proof of Proposition 9. As in the proof of Proposition 1, the first order conditions of the optimization problem can be written as:

u (c 0,t ) -θR t+1 p1 p1 u (c 1,t+1 ) = µ t , u (c 0,t ) -θ p1 while the complementary slackness conditions are given by (28). By rearranging equations in system (60), we obtain:

where R t+1 is defined in [START_REF] Cremer | Collective annuities and redistribution[END_REF]. The rest of the proof is similar to the one of Proposition 1.


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

The 90% public debt threshold: The rise and fall of a stylised fact

Keywords: JEL classification codes: E6, F3, F4, N4 public debt, economic growth, nonlinearity, threshold effects 1

, descriptive statistics reveal that real GDP growth slows considerably as the central government debt-to-GDP ratio goes beyond the 30% threshold and that no further slowdown can be observed in the data as the debt-to-GDP ratio rises above 60% and 90% during the periods 1790-2009 and 1946-2009. For the United States (1946-2009), the negative nonlinear finding completely disappears for any level of public debt, once reverse causality and influential outliers are accounted for. Looking at general (and central) government debt during the more recent period of 1960-2009 suggests that economic slowdown occurs when public debt moves above 60% or 90% of GDP. But it seems more appropriate to determine nonlinearity and the associated debt threshold endogenously. Therefore, in a second stage, we put the Reinhart-Rogoff dataset to a formal econometric test by employing nonlinear threshold models. Overall, our estimation results indicate that the nonlinear relation from debt to growth is not very robust. Taken with a pinch of salt, our results suggest, however, that there may be a tipping point at around 20% of GDP, beyond which central government debt has a negative influence on growth. Further (and greater) thresholds may exist but their magnitude is highly uncertain. For general government debt , the threshold beyond which negative growth effects kick in is considerably higher at about 50%. Finally, individual country estimates reveal a large amount of cross-country heterogeneity. For some countries including the United States, a nonlinear negative link can be detected at about 30% of GDP. For others, the thresholds are surrounded by a great amount of uncertainty or no nonlinearities can be established. This instability may be a result of threshold effects changing over time within countries and depending on economic conditions, not captured in our estimations. Overall, our results can be seen as a formal econometric confirmation that the 90% public debt threshold is not in the data. But our results also seem to suggest that public debt might have a negative effect on economic performance kicking in at already fairly moderate public debt levels. Furthermore, the absence of threshold effects or low estimated thresholds may not preclude the emergence of further threshold effects, especially as public debt levels are rising to unprecedentedly high levels.

Introduction

Using simple descriptive statistics, [START_REF] Reinhart | Growth in a time of debt[END_REF] argued for the existence of strong negative effects of high public debt on economic growth. In particular, they pointed out that economic growth slows down considerably if the public debt-to-GDP ratio exceeds 90%. But some of the calculations presented in [START_REF] Reinhart | Growth in a time of debt[END_REF] seem to be flawed. [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF] tried to reproduce the Reinhart-Rogoff descriptive statistics and uncovered simple spreadsheet errors and irregularities with regard to country and time coverage. As a result, the sharp drop in real GDP growth above the 90% debt threshold becomes a mild decrease and the strong conclusion, implied by the Reinhart and Rogoff finding, often used in the policy debate, that countries had better avoid the 90% debt threshold seems less obvious.

Many empirical papers published in the aftermath of [START_REF] Reinhart | Growth in a time of debt[END_REF] validated the debt 90% threshold. For instance, [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.

Yet, a new wave of papers started casting doubt on the one-size-fits-all feature of the 90% debt threshold. [START_REF] Caner | Finding the tipping point -when 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. [START_REF] Baglan | Government debt and macroeconomic activity: a predictive analysis for advanced economies[END_REF] identify a threshold effect of 20% of GDP for low-debt countries and a negative linear relationship between debt and growth for high-debt countries. [START_REF] Minea | Is high public debt always harmful to economic growth? Reinhart and Rogoff and some complex nonlinearities[END_REF] find a debt threshold at 115% of GDP. [START_REF] Panizza | Public debt and economic growth: is there a causal effect?[END_REF] even 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.

Using a variant of the Reinhart and Rogoff dataset, Égert (2012) demonstrated that the negative nonlinear relationship between debt and growth is very sensitive to empirical modelling choices. But even if you take the presence of such threshold effects as a given, they kick in at much lower levels of public debt (between 20% and 60% of GDP). This paper seeks to take this analysis a step further by putting the original Reinhart-Rogoff dataset used in [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF], to a formal econometric testing. We attempt to identify 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 periods 1790 to 2009 and 1946 to 2009. Furthermore, we discuss the implications of data definitions. The Reinhart and Rogoff dataset comprises central government debt. Yet more relevant for policy discussion is the general government debt, ie the consolidated debt of all levels of government included central government, social security administrations and subnational governments. The difference between the two public debt series can be sometimes very large. Finally, we also carry out country specific estimations on the Reinhart and Rogoff dataset to see whether the relationship between public debt and growth varies across countries.

The paper is organised as follows. Section 2 provides descriptive statistics of and a first visual glance at the Reinhart-Rogoff dataset. Section 3 presents the estimation strategy. Section 4 reports and discusses the empirical results. Section 5 finally summarises and provides some policy conclusions. [START_REF] Reinhart | Growth in a time of debt[END_REF] use descriptive statistics to show the detrimental effect of high levels of public debt on real GDP growth. They show for a group of twenty OECD countries and for 1946 to 2009 that average GDP growth drops from more than 3% to zero as the public debt-to-GDP ratio increases above 90%. But according to [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF], the average annual growth is 1.9%, and not -0.1%, when public debt is above 90% of GDP (Table 1). Using a variant of the Reinhart and Rogoff dataset 2 , Égert (2012) also found that the dramatic drop in real GDP growth did not occur above the 90% threshold (Table 1). In this paper, we use the Reinhart-Rogoff dataset made public by [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF] 3 and our results are, unsurprisingly, in line with those reported in [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF] 4 .

Stylised facts

While the averages computed by [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF] do not indicate a negative real GDP growth for debt levels above 90%, their number still suggest that economic growth slows down as one moves above the 90% debt threshold. But annual data may be just too noisy to reveal us the true picture. We therefore computed 10-year non-overlapping averages for real GDP growth. The average of these multiyear averages, reported in Table 1, show that GDP growth does not slow down at high levels of public debt. In fact, there seems to be a deceleration only if public debt exceeds 30% of GDP. By contrast, GDP growth remains stable, or even slightly increases as public debt increases further. This observation remains valid for 5-or 8-year averages as well. 5But as argued by many observers, whether causality runs from debt to growth or whether lower growth results in higher debt remains to be seen. An easy way to tackle this problem is to compare average GDP growth with past debt levels. Looking at average annual growth rates indicates that there is no economic slowdown beyond the 90% debt ceiling. In fact, GDP growth decreases from 4% to 3% as public debt increases from below 30% to between 30% and 60%. Nevertheless, GDP growth remains stable at 3% as public debt moves above 60% and 90% of GDP. This finding holds equally for multiyear average growth rates.

2 Égert (2012) matched data on central government debt obtained from the data appendix of [START_REF] Reinhart | From financial crash to debt crisis[END_REF] with real GDP growth rates available from the Barro-Ursúa macroeconomic dataset (Barro and Ursúa, 2011). The difference with the actual Reinhart and Rogoff data is that Égert (2012)'s data excludes Ireland and includes Switzerland and that the data series used in Égert (2012) are longer.

3 Available here: http://www.peri.umass.edu/236/hash/31e2ff374b6377b2ddec04deaa6388b1/publication/566/ 4 There are three differences in the dataset used in the paper and the one used by [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF]. First, for France, the public debt series has a sudden break in 1978-1979 (8.8% in 1978 and 31.1% in 1979). We decided to use the series calculated from the separate nominal debt and nominal GDP series provided by [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF]. This leaves us with missing values from 1973 to 1977 but we avoid the abrupt break, probably due to a change in methodology/definition. Another difference is that [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF] use the Greek public debt to GDP ratio starting in 1970, while the series can be computed from 1948 onwards. Finally, [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF] exclude 1956 for the Netherlands, even though the observation is not missing from their background data file. The Reinhart-Rogoff dataset allows the assessment of more than 200 years for some countries. Annual average real GDP growth rates computed for 1790-2009 confirm the possibility of a negative correlation between central government debt and economic growth: GDP growth decreases steadily from 4% to 2% as government debt rises from below 30% to above 90% of GDP. But when looking at the relation between growth and lagged central government debt (in order to control for reverse causality), growth drops from about 4% to below 3% with debt exceeding 60% of GDP, but no further decline can be observed beyond 90% of GDP. For the period 1790 to 1939, Table 2 also shows that growth slows down above debt levels exceeding 30% but that growth accelerates mildly if debt is higher than 90%. [START_REF] Reinhart | Growth in a time of debt[END_REF] argue that the 90% debt threshold can be observed for the US economy: public debt exceeding the threshold of 90% goes in tandem with a decline in annual growth from about 3.5% to well below zero (Table 3). Our data replicates fairly well this finding. Nevertheless, this result is largely influenced by one single influential outlier, real GDP growth of -11% in 1946. This is well demonstrated in Table 3: when reducing the sample by one year from 1946-2009 to 1947-2009, average real GDP growth in the 90% and higher debt regime changes from -2% to 1%. Moving one step further and comparing GDP growth rates with central government debt a year earlier eliminates completely the finding that GDP growth slows down if debt goes above 90% of GDP. In fact, average real GDP growth rates are very close to 3% irrespective of the level of central government debt. More generally, it does not seem to make too much sense to compute averages for the 90%+ debt regime, considering that, for the period 1946 to 2009, there are only four years (and four observations) falling into this regime (1946 to 1949), including the influential outlier in 1946. [START_REF] Reinhart | Growth in a time of debt[END_REF] and [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF] are focusing on central government debt. Nevertheless, drawing far-reaching policy conclusions based on central government debt may be misleading. In fact, what matters for public debt is the consolidated debt of the government sector (general government debt), which includes not only central government but also social security administrations and subnational governments. As Figure 1 hereafter shows, the average difference is a little higher than 20 percentage points but it can go as far as 50 percentage points for Canada. For this reason, we also calculate averages using general government debt obtained from the OECD's Economic Outlook 92 database. Data on general government start in 1960 at the earliest but for some countries (Greece, Ireland, Portugal), they only start in the mid-1990s. 6 For the sake of comparability, average GDP growth rates are also calculated for central government data matching exactly the time span of the general government debt series.

As a matter of fact, average real GDP growth rates decline gently with general government debt increasing up to 90%, but it then drops from 3% to 2% as debt goes beyond 90% of GDP (Table 4). This important decline is confirmed when looking at GDP growth rates as a function of the (one year) lagged general government debt-to-GDP ratio but the smooth decline disappears at lower debt levels, where growth seems to be unrelated to the level of general government debt. A similar pattern can be observed when using multiyear averages. A smooth decline in growth rates occur when comparing debt and growth. But if we use lagged debt, the one percentage point slowdown in economic growth is evenly spread when moving from 30%-60% to above 90% of GDP. Let us now look at average growth rates in function of central government debt. Annual averages suggest a gradual slowdown in growth while multiyear averages indicate that growth decelerates considerably as central government debt goes beyond 60% of GDP (and not 90%).

Figure 1. The difference between general government debt (%of GDP) and central government debt (%of GDP), 2009 Percentage point

Source: Author's calculations using general government debt obtained from the OECD Economic Outlook 92 database and central government debt from the Reinhart-Rogoff dataset. Another way of having a preliminary idea about the correlation between growth and debt is to plot the annual GDP growth rates against the debt ratio. The scatter plots presented in Panel A of Figure 2 are striking: annual GDP growth and the central government debt ratio do not appear to have any apparent relationship with one another for the sub-periods considered, perhaps with the exception of the period 1946-2009 where a little kink can be seen for low debt levels. The general picture does not change if public debt is plotted with a lag of one year (Panel B of Figure 2). Visual inspection yields a similar general impression for growth and general government for 1960 to 2009 (Panel C of Figure 2) and for the US economy (Figure 3).

Multi-year averages eliminate cyclical and other short-term effects, which may contaminate the scatter plots of annual figures. Therefore, Figure 4 plots non-overlapping 10-year averages for growth and central government debt for the period 1946-2009 (Reinhart-Rogoff data) and for general government debt for 1960 and 2009. Eyeball econometrics does suggest some kind of linear negative correlation between debt and growth, but no apparent threshold effects.

Econometric issues

We estimate bivariate threshold models, in which the effect of debt on growth depends on the level of debt. We use the testing procedure developed by [START_REF] Hansen | Threshold Effects in Non-Dynamic Panels: Estimation, Testing and Inference[END_REF], which helps determine the threshold values endogenously through a grid search and which 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 (1a)                  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          (1b)
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 minimises 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. But we also experiment with alternative paramtetrisation (30%, 10%, 5% and 1%).

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 two-regime 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 10% 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 is 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 two-regime and three-regime models.

Estimation results

Central government debt and the Reinhart-Rogoff dataset

Reinhart and Rogoff (2010) imposed the 90% debt threshold without any formal testing. But it seems more appropriate, as argued earlier, to test empirically whether there are debt thresholds in the data and if so, where they are located. Before diving into the details, we have to emphasise that 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. Using the lagged public debt-to-GDP ratio helps circumvent this problem: in our bivariate setup, it is almost certain that lagged debt may have an influence on growth but not the other way around. Therefore, we focus on the interpretation of this relationship (lagged debt and growth) in what follows.7 Our estimation results indicate that there is a negative nonlinear relationship between (lagged) central government debt and growth for the period 1790 to 2009 (Table 5). But there is uncertainty whether this nonlinear relationship includes two or three different regimes and where the debt thresholds are. Depending on the minimum number of observations required to be included in the outer regimes: the results indicate a two-regime model with a threshold at about 30% of GDP if a large number of observations are included in one regime (30% of the observations) and a three-regime model with thresholds of 4% and 90%, if the minimum number of observations is 1% of total observations. Obviously, the lower the minimum number of observations in specific regimes, the higher the probability that a very low or very high threshold will be picked. But at the same time, the results may be less general because they will be more sensitive to outliers.

The coefficient estimates are negative in the high-debt regimes but they tend to be lower than the negative coefficients obtained for lower debt regimes. 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 therefore re-run the estimations using the (lagged) rate of growth of central government debt rather than the (lagged) level of the debt-to-GDP ratio as independent (nonlinear) variable and using, as before, the lagged public debt-to-GDP ratio as the threshold variable. The new results seem to be more stable. Always the three-regime model is selected, with a lower threshold of around 20% of GDP and an upper threshold of roughly 60% of GDP. The coefficient estimates indicate that public debt higher than 20% of GDP starts having a negative impact on growth. But the negative coefficient in the upper debt regime remains lower compared to the one in the middle debt regime, which is inconsistent with the Reinhart-Rogoff claim. An exception is the case when only 1% of total observations is required to be in the outer regimes: the upper threshold moves to 150% of GDP and the negative coefficient becomes massively negative in the high debt regime.

Results obtained for the period 1946-2009 are broadly in line with the earlier results (Table 6). The negative effect of central government debt on growth kicks in at about 20% of GDP. In some cases, there is another debt threshold at about 60% of GDP, but the coefficients above this threshold are lower than below it. We carry out the estimations again using the growth rate of public debt as a nonlinear variable. The results indicate the presence of a 20% debt threshold above which a one percent change in central government debt reduces growth by 0.04 percentage points. Furthermore, there is some but not very robust evidence for another debt threshold which is somewhere between 55% and 130% of GDP, beyond which the negative impact on growth grow much stronger. 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].

Whether debt will have a nonlinear negative effect on growth, and at what level, depends on the importance of the interest rate channel. The overall negative impact will be given by the combination of the three channels. Against this backdrop, here we set out to assess possible country-specific nonlinearities by estimating threshold models for individual countries using the Reinhart and Rogoff dataset of central government debt.

The results, summarised in Table 8, highlight several country specificities. First of all, a robust nonlinear negative relationship linking public debt and growth can be established only for a handful of countries including Belgium, Finland, Germany and the United States. The debt threshold beyond which the negative impact of debt on growth kicks in is very low, around or even below 30% of GDP. But the negative effects are very different: relatively small for Belgium but more important for Germany and the US economy. Second, for another subgroup of countries including Austria, Canada and Ireland, there is a large degree of uncertainty around the tipping point, which ranges somewhere between 30% and 70%, as it is sensitive to modelling choices, namely to the minimum number of observations included in one regime (10% vs. 20%). Third, there is no nonlinear relationship between public debt and growth in some countries such as Australia and Spain. In these two countries, there is not even a negative linear link between debt and growth. Fourth, in some other countries like Denmark, Italy and Japan, even though the presence of nonlinearity can be detected, the relation between debt and growth is positive in the high debt regime. Finally, in the remaining countries, whether public debt has a negative or positive influence on real GDP growth above a certain level of the central government debt-to-GDP ratio depends on the minimum number of observations required for individual regimes. threshold and that no further slowdown can be observed in the data as the debt-to-GDP ratio rises above 60% and 90% during the periods 1790-2009and 1946-2009. For the United States (1946-2009), the negative nonlinear finding completely disappears for any level of public debt, once reverse causality and influential outliers are accounted for. Looking at general (and central) government debt during the more recent period of 1960-2009 suggests that economic slowdown occurs when public debt moves above 60% or 90% of GDP.

Given that it is more appropriate to determine possible threshold effects in an endogenous fashion, in a second stage, we put the Reinhart-Rogoff dataset to a formal econometric test by employing nonlinear threshold models to investigate the presence of a possible negative nonlinear relationship between debt and growth. Overall, our estimation results indicate that the nonlinear relation from debt to growth is not very robust. Taken with a pinch of salt, our results suggest, however, that there may be a tipping point at around 20% of GDP, beyond which debt has a negative influence on growth. Further (and greater) thresholds may exist but their magnitude is highly uncertain. For general government debt , the threshold beyond which negative growth effects kick in is considerably higher at about 50%. Finally, individual country estimates reveal a large amount of cross-country heterogeneity. For some countries such as Germany and the United States, a nonlinear negative link can be detected at about 30% of GDP. While negative nonlinearities, surround by a huge amount of uncertainty regarding the quantitative effects, seem to be present in some, no robust or absolutely no negative nonlinearities can be established in a number of countries. This instability may be a result of nonlinear effects changing over time within countries and economic conditions.

Our results can be seen as a formal econometric confirmation that the 90% public debt beyond which economic growth slows significantly is not in the data: the previous stylised fact is indeed a statistical fallacy. But our results also seem to suggest that public debt might have a negative effect on economic performance kicking in at fairly moderate public debt levels. Nevertheless, the magnitude of those effects and the precise size of the thresholds may vary to a great extent across countries. Furthermore, the absence of threshold effects or low estimated thresholds may not preclude the emergence of further threshold effects, especially as public debt levels are rising to unprecedentedly high levels.

Table B6. Country-specific results

Minimum % of observations required in one regime: 20%

  Figure 2. Public debt (%of GDP) and real GDP growth, annual dataPanel A -Reinhart-Rogoff dataset, contemporaneous central government debt1790-2009 1790-1939 1946-2009

Table 1 . Real GDP growth and central government debt as a % of GDP, 1946-2009 Level of central government debt (as a % of GDP) x<30% 30%< x<60% 60%<x<90% x>90% Reinhart-Rogoff (2010) Average annual growth rates

 1

	3.9	2.9	3.5	-0.1

level of central government debt x<30% 30%< x<60% 60%<x<90% x>90% Average of annual growth rates

		4.1	3.1	3.1	2.9
	Average of 10-year average
	growth rates	3.6	2.6	2.8	2.7

Table 2 . Real GDP growth and central government debt as a % of GDP, 1790-2009 Level of central government debt (as a % of GDP) x<30% 30%< x<60% 60%<x<90% x>90% Level of central government debt (as a % of GDP)

 2

	1790-2010	4.0	3.1	2.5	2.2
	1790-1939	3.7	2.9	1.9	2.3

Level of lagged central government debt (as a % of GDP)

  Source: author's calculations based on the Reinhart-Rogoff dataset

	1790-2010	3.7	3.0	2.6	2.7
	1790-1939	3.2	2.7	2.1	2.6

Table 3 . Real GDP growth and central government debt as a % of GDP, United States

 3 Source: author's calculations based on the Reinhart-Rogoff dataset

		period	x<30% 30%< x<60% 60%<x<90% x>90%
			central government debt (as a % of GDP)
	Reinhart-Rogoff (2010) 1790-2009	4.0	3.4	3.3	-1.8
	This paper	1790-2009	4.1	3.2	3.3	-2.0
			lagged central government debt (as a % of GDP)
		1790-2009	3.7	4.1	3.2	2.9
			central government debt (as a % of GDP)
		1946-2009	Na	3.4	3.3	-2.0
		1947-2009	Na	3.4	3.3	1.0
			lagged central government debt (as a % of GDP)
		1946-2009	Na	3.3	3.2	2.9

Table 4 . Real GDP growth and general (and central) government debt as a % of GDP, 1960-2009 x<30% 30%< x<60% 60%<x<90% x>90% Average annual real GDP growth rate

 4

	General government debt	3.3	3.0	2.8	1.9
	Lagged general government debt	2.8	2.9	2.9	2.0
	Central government debt	3.3	2.8	2.3	2.0
	Lagged central government debt	3.0	2.8	2.6	2.2
		Average of 10-year average real GDP growth rates
	General government debt	3.6	3.3	2.8	2.1
	Lagged general government debt	2.8	2.8	2.2	1.9
	Central government debt	3.5	3.1	2.1	2.1
	Lagged central government debt	2.7	2.7	1.7	2.0

Table 5 . Reinhart-Rogoff dataset, 1790-2009 Minimum % of observations required in one regime

 5

			30%	20%	10%	5%	1%
			Nonlinear variable = lagged central government debt/GDP
			Threshold variable = lagged central government debt/GDP
	Test of nonlinearity				Bootstrapped p-value
	H0: linear vs. H1: 2-regimes	0.084	0.000	0.000	0.000	0.000
	H0: 2-regimes vs. H1: 3-regimes	0.184	0.248	0.060	0.082	0.054
	Coefficients	Low debt	0.015	-0.022**	0.044	0.044	0.699
		Middle debt			-0.018**	-0.018**	-0.020**
		High debt	-0.006	-0.009**	-0.006*	-0.006*	-0.007**
	Debt thresholds (%) Threshold 1	27.72	71.99	14.27	14.27	4.40
		Threshold 2			94.27	94.27	94.27
	No. of OBS		2177	2177	2177	2177	2177
			Nonlinear variable =

lagged rate of growth in central government debt/GDP Threshold variable = lagged central government debt/GDP

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

	Test of nonlinearity				Bootstrapped p-value
	H0: linear vs. H1: 2-regimes	0.002	0.004	0.000	0.002	0.002
	H0: 2-regimes vs. H1: 3-regimes	0.004	0.002	0.000	0.002	0.000
	Coefficients	Low debt	0.003	0.004	0.004	0.004	0.004
		Middle debt	-0.041**	-0.038**	-0.038**	-0.038**	-0.010**
		High debt	-0.007**	-0.007**	-0.007**	-0.007**	-0.242**
	Debt thresholds (%) Threshold 1	23.64	19.62	19.62	19.62	13.48
		Threshold 2	52.98	67.86	67.86	67.86	155.00
	No. of OBS		2120	2120	2120	2120	2120
	Note:

Table 6 . Reinhart-Rogoff dataset, 1946-2009, annual data

 6

		Minimum % of observations required in one regime
	30%	20%	10%	5%	1%

Table 8 . Country-specific results, 1790-2009

 8

		Nonlinear variable = lagged growth rate of central government debt/GDP
			Threshold variable = lagged central government debt/GDP
		Minimum % of observations required in one regime: 20%
		Test of nonlinearity (p-value)		Coefficients		Debt thresholds (%) No obs
		lin vs 2reg	2reg vs. 3reg	low	middle	high	Low	High
	AUS	0.28	0.42	0.068*
	AUT	0.00	0.02	0.027	0.093	-0.236**	18.94	58.15
	BEL	0.02	0.46	0.061		-0.021*	29.36
	CAN	0.00	0.00	0.096	-0.240** -0.605**	45.36	55.42	74
	DEU	0.00	0.00	0.074**	-0.002	-0.159**	13.43	24.09	57
	DNK	0.00	1.00	-0.037		0.056**	40.53		83
	ESP	0.55	0.29	0.001
	FIN	0.00	0.00	0.020	-0.013	-0.113**	12.02	16.96	80
	FRA	0.00	0.00	-0.060** -0.006**	0.086**	35.33	70.44	92
	GBR	0.33	0.11	0.036
	GRC	0.00	0.00	0.059**	-0.270**	-0.003	24.57	110.74
	IRL	0.00	1.00	-0.208**		-0.133**	27.84		62
	ITA	0.00	0.13	-0.063		0.068**	28.58
	JPN	0.00	0.00	0.116*	0.062	-0.053	19.41	54.08	96
	NLD	0.01	0.00	-0.072**	0.075	-0.187**	47.87	70.69
	NOR	0.00	0.00	-0.023	-0.022	0.087**	21.57	26.99
	NZL	0.00	0.00	0.108	-0.064	-0.241**	37.11	53.45	67
	PRT	0.00	0.00	-0.018	-0.085	0.082	54.27	70.12	87
	SWE	0.00	1.00	0.099**		0.002	18.88
	USA	0.00	0.03	0.000	0.035	-0.159**	17.25	33.73

Nonlinear variable = lagged central government debt/GDP Threshold variable = lagged central government debt/GDP Test of nonlinearity (p-value)

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

					Coefficients		Debt thresholds (%) No obs
		lin vs 2reg	2reg vs. 3reg	low	middle	high	Low	High
	AUS	0.12	0.35	-0.007					143
	AUT	0.00	0.00	0.489**	0.046*	0.153*	15.24	36.45	107
	BEL	0.11	0.07	0.010					156
	CAN	0.00	0.00	0.059	0.038	0.086*	44.44	53.64	76
	DEU	0.00	0.00	0.061	-0.068	0.258*	8.36	11.54	58
	DNK	0.00	1.00	0.079*		0.001	21.82		86
	ESP	0.00	0.16	0.102**		0.018	51.89		131
	FIN	0.00	0.00	0.031	0.038	0.178*	12.02	26.43	82
	FRA	0.00	0.00	0.335**	0.029*	0.074**	25.67	81.45	96
	GBR	0.00	0.00	-0.056**	-0.008	-0.026**	43.23	135.69	164
	GRC	0.00	0.00	0.341**	0.003	0.036*	22.43	121.53	109
	IRL	0.00	1.00	-0.191**		-0.031*	27.84		64
	ITA	0.00	0.63	0.206**		0.010	32.78		111
	JPN	0.00	0.00	0.096	-0.040** -0.121**	16.83	48.03	99
	NLD	0.00	1.00	-0.064**		-0.032**	50.36		103
	NOR	0.00	1.00	0.283**		0.201**	24.53		115
	NZL	0.00	1.00	0.002		0.029*	58.03		69
	PRT	0.00	0.00	0.267**	0.004	0.063	18.31	54.27	89
	SWE	0.00	0.02	0.209**	0.045*	0.146**	16.89	35.26	115
	USA	0.02	0.36	-0.119*		-0.021*	22.17		204
	Note:

Table B7 . Country-specific results

 B7 Minimum % of observations required in one regime: 10%

Nonlinear variable = central government debt/GDP Threshold variable = central government debt/GDP Test of nonlinearity (p-value)

				Coefficients		Debt thresholds (%) No obs
	lin vs 2reg	2reg vs. 3reg	low	middle	high	Low	High

These results are not reported here but can be obtained upon request from the author.

The data appendix provides more details on the time coverage.

Appendix B reports results for the relationship between contemporaneous public debt and growth. The results do not differ too much from those obtained using lagged public debt.

Nonlinear variable = lagged central government debt/GDP Threshold variable = lagged central government debt/GDP

General government debt

We repeat the above exercise replacing central government debt by general government debt, a more relevant measure for policymakers. The tests of nonlinearity indicate that the null hypothesis of a linear model cannot be accepted against the alternative of a two-regime model, and sometimes even a three-regime model is selected over a two-regime model (Table 7). The estimated thresholds appear to be sensitive to the parametrisation of the threshold models 12 (minimum number of observations required in a specific regime) and range from 50% to 90% of GDP when lagged general government debt is the nonlinear variable. By contrast, the results are more straightforward if the rate of growth of the general government debt ratio is taken as the nonlinear variable. In that case, the significant negative impact of public debt on growth becomes visible if debt exceeds 45-50% of GDP: a one percent growth in debt reduces growth by almost 0.1 percentage points. These results are broadly confirmed by estimations performed on central government debt data even though the degree of uncertainty is much greater (Table B4 in Appendix B): a tipping point lies somewhere in the range of 30% to 70% beyond which a one percent change in central government debt decreases growth by 0.04 -0.12 percentage points.

Individual country estimates

Thus far, we have assumed that the debt-growth relationship is homogenous across countries: the same slope coefficients and debt thresholds are assumed to hold for the 20 countries included in the Reinhart-Rogoff dataset. Yet this assumption may be too restrictive given that public debt can affect economic growth differently in different countries. There are a number of channels through which public debt is likely to hamper long-term growth. They are as follows:

 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.

Conclusions

The aim of this paper was to contribute to the empirical literature on the identification of a possible debt threshold beyond which negative effects for economic growth appear. We analyse the original Reinhart-Rogoff dataset, made public by [START_REF] Herndon | Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff[END_REF], relying on descriptive statistics and formal econometric testing. First, employing the debt thresholds (30%, 60% and 90%) proposed by [START_REF] Reinhart | Growth in a time of debt[END_REF], we show using descriptive statistics that real GDP growth slows considerably as the central government debt-to-GDP ratio goes beyond the 30% Appendix A