Jean-Baptiste Michau
email: baptiste.michau@polytechnique.edu

Secular Stagnation: Theory and Remedies

Keywords: Liquidity trap, Monetary and …scal policy, Secular stagnation JEL Classi…cation: E12, E31, E62, E63

This paper relies on a Ramsey model with money to o¤er a simple theory of secular stagnation. The permanent failure of the economy to produce at full capacity results from three features: (i) The combination of the zero lower bound on the nominal interest rate and of an in ‡ation ceiling imposes a lower bound on the real interest rate; (ii) Some dynastic households have a high propensity to save, due to a preference for wealth; (iii) A downward wage rigidity breaks the de ‡ationary spiral resulting from the lack of demand. In this framework, I derive the paradox of ‡exibility, of thrift, and of toil.

If the in ‡ation ceiling cannot be raised, then the government needs to rely on …scal policy to escape secular stagnation. However, a conventional …scal stimulus is not an e¢ cient response to a permanent liquidity trap, and can even be welfare reducing. The solution is instead to tax household wealth and to subsidize income from physical capital, through an investment subsidy or a reduction in the taxation of corporate income. This optimal policy is revenue neutral and implements the …rst-best allocation of resources. However, to avoid a jump in the price level upon implementation of the optimal policy, the government needs to redeem the money that had previously been supplied to …nance public de…cits.

Introduction

depressions are usually seen as an extreme manifestation of a business cycle phenomenon, Hansen (1939) was the …rst to worry that it might be a permanent state of a¤airs. This is the "secular stagnation" hypothesis.

When aggregate demand is weak, in ‡ation declines and the central bank responds by aggressively cutting the nominal interest rate such as to stimulate demand. However, a fundamental constraint on monetary policy is that the nominal interest rate cannot be negative. Indeed, no-one is willing to pay more than $100 for a bond yielding $100 in the future. In the mid-1990s, Japan was the …rst large industrialized country to hit the zero lower bound and to fall into the liquidity trap. Its subsequent history, with an economy mired in very low in ‡ation and weak GDP growth, suggests that there is no mechanism through which an economy naturally recovers from a persistent lack of demand. Moreover, Japan is facing a policy conundrum as aggregate demand remains desperately weak despite a budget de…cit of about 7% of GDP, a debt-to-GDP ratio of 230%, and substantial monetary accommodation.

Over the past few years, there has been growing concerns that the U.S. and the Eurozone might be in a similar situation as Japan. In particular, Summers (2014) has emphasized that the credit boom of 2003-2007 in the U.S. did not generate a corresponding economic boom, which suggests that the unsustainable demand for consumption of poor borrowers was hiding an already weak level of demand of rich lenders.

Despite the prominence of the secular stagnation hypothesis in the policy debate, there has been few attempts to model it explicitly. In this paper, I o¤er a simple theory of secular stagnation, on which I then rely for a careful policy analysis. The structure of the model is a Ramsey economy with money and ‡exible prices, to which I add three features.

First, I assume that the central bank imposes a ceiling on in ‡ation. This, together with the zero lower bound on the nominal interest rate, generates a lower bound on the real interest rate. In most industrialized countries, central banks never allow in ‡ation to exceed 2%, which prevents the real interest rate from ever falling below -2%.

Second, I assume that households derive utility from holding wealth. This can raise their propensity to save to such an extent that a negative real interest rate is necessary for the economy to produce at full capacity. Importantly, in…nitely lived households should be interpreted as dynasties. Hence, bequest motives are important determinants of the saving behavior of households. In particular, we know that pure altruism alone cannot account for the observed patterns of bequests (Kopczuk 2009). Instead, parents seem to be commonly motivated by the "joy of giving". A "capitalist spirit" also induces many individuals to derive intrinsic utility from the accumulation of wealth, which is then passed on from generation to generation (Carroll 2000, Kopczuk 2007). [START_REF] Acemoglu | Introduction to Modern Economic Growth[END_REF] While I rely for simplicity on a representative household model, my results only require that a strictly positive mass of households has a preference for wealth. Indeed, a well known result in macroeconomics is that the real interest must eventually be determined by the behavior of the most patient households. This is highly relevant as the rise in inequality is often cited as a major cause of the recent decline in aggregate demand (Summers 2014). A preference for wealth seems a natural explanation for the concentration of wealth in the hands of a small number of households with a very low propensity to spend (Kumhof, Rancière and Winant 2015).

Finally, I impose a downward wage rigidity. If the in ‡ation ceiling is su¢ ciently low and the preference for wealth su¢ ciently strong, then aggregate demand is smaller than aggregate supply. This reduces in ‡ation, which raises the real interest rate, which further contracts aggregate demand. To have a stationary secular stagnation equilibrium, I therefore need to put a break on the de ‡ationary spiral. This is achieved through the downward wage rigidity.

In the secular stagnation equilibrium, aggregate demand is depressed. This induces …rms'labor demand to fall short of workers'labor supply. However, this ine¢ ciency is primarily due to the lower bound on the real interest rate, not to the downward wage rigidity. Indeed, if wages are more ‡exible, then in ‡ation is smaller, the real interest is higher, and the economy is even more depressed. Conversely, for a su¢ ciently high in ‡ation ceiling, there always exists a frictionless steady state equilibrium where the economy produces at full capacity. [START_REF] Benhabib | The Perils of Taylor Rules[END_REF] The obvious policy response to secular stagnation is to raise the in ‡ation ceiling. However, in most countries, this seems politically and institutionally out of reach. Hence, I explore alternative remedies. A simple policy analysis con…rms that the usual set of tools used for macroeconomic stabilization are not adequate in the context of secular stagnation. Increasing the money supply through open market operations is useless in a liquidity trap. A helicopter drop of money that is su¢ ciently large to induce the economy to produce at full capacity is inconsistent with the in ‡ation ceiling. A …scal stimulus raises …rms'demand for labor much more than workers'consumption level. Hence, it is likely to be welfare reducing, even though the government spending multiplier is typically well above 1.

However, with a wider set of policy instruments, the government can implement the …rst-best allocation of resources without raising the in ‡ation ceiling. To understand this result, note that secular stagnation is fundamentally due to an excessively high real income ratio is a rising function of r g, consistently with the historical experience of France from 1820 to 2009. interest rate. This creates two distortions. First, it raises the cost of capital. This reduces the demand for investment, resulting in depressed capital-labor ratio. Second, a high real interest rate reduces the demand for consumption. The …rst distortion should be addressed by subsidizing the income from physical capital. This can either be achieved through an investment subsidy or a reduction in the taxation of corporate income. The second distortion can be o¤set through a wealth tax, which reduces the e¤ective real interest rate faced by consumers. This policy is revenue neutral and it exactly replicates the allocation of resources that would result from an increase in the in ‡ation ceiling.

By inducing the economy to produce at full capacity, the optimal policy makes the price level proportional to the money supply. Hence, to prevent a jump in the price level upon implementation of the policy, the government must redeem the money supply that is not used for transactions. This suggests that relying on the printing press to …nance some stimulus can eventually make the government reluctant to implement the optimal policy. Related Literature. Secular stagnation is a stationary phenomenon. However, in typical models of the liquidity trap, the binding zero lower bound forces the real interest rate to be above households' discount rate (Krugman 1998, Eggertsson and Woodford 2003, Werning 2012). This induces households to adopt a rising path of consumption, which is inconsistent with stationarity. A model of secular stagnation must therefore reconcile an excessively high real interest rate, which depresses aggregate demand, with the existence of a stationary equilibrium. Eggertsson and Mehrotra (2014) solved the problem by relying on an overlapping generation model without altruistic links. In their theory of secular stagnation, aggregate demand is either depressed because of a low rate of population growth or because the economy has been hit by a deleveraging shock. [START_REF] Benigno | Stagnation Traps[END_REF] Interestingly, to eliminate the output gap, the government can either increase its spending or redistribute income from lenders to borrowers. Kocherlakota (2013) showed that, with overlapping generations or credit constraints, secular stagnation can result from a fall in the price of land. Caballero and Farhi (2015) relied on a perpetual youth model with risky assets to o¤er an alternative theory of secular stagnation where the lack of demand is due to a shortage of safe assets. To escape the depression, the government should either allow in ‡ation to increase or raise the supply of public debt, possibly by buying private risky assets.

My contribution to this literature is to show that secular stagnation can be derived in a Ramsey economy where some households have a preference for wealth. Under a dynastic interpretation of in…nitely lived households, my work shows that altruistic links across generations are not inconsistent with the existence of a stationary secular stagnation equilibrium. On the contrary, the bequest motive seems to be a natural justi…cation for the preference for wealth that delivers stationarity. The Ramsey framework is convenient for policy analysis. This allows me to characterize the optimal policy that implements the …rst-best allocation of resources without raising the in ‡ation ceiling. Also, by introducing money into the economy, I can investigate the monetary …nancing of public de…cits.

In a related contribution, Michaillat and Saez (2015) o¤er a model of the business cycle where self-employed households exchange their labor services under matching frictions. As prices and market tightness are not independently determined, they focus on an equilibrium where in ‡ation is constant and where market tightness adjusts such as to equate aggregate demand to aggregate supply. Hence, business cycle ‡uctuations result in ine¢ cient ‡uctuations in market tightness. To have a permanent liquidity trap, they also assume that households have a preference for wealth. Interestingly, as a lack of demand does not translate into lower in ‡ation, they do not obtain the Keynesian paradoxes.

Another strand of the literature has emphasized the role of expectations. Benhabib, Schmitt-Grohé and Uribe (2001) showed that, if the central bank follows a Taylor rule, then self-ful…lling expectations can induce the economy to fall into a permanent liquidity trap. Benigno and Fornaro (2015) proposed a theory of "stagnation traps" where the weakness of aggregate demand depresses investment in innovation, resulting in a low rate of technological progress. This pushes the nominal interest rate against the zero lower bound, which explains why aggregate demand remains so weak. In that context, subsidies to investment in innovation are welfare enhancing.

The optimal policy that I derive is similar to the "unconventional …scal policy" of Correia, Farhi, Nicolini and Teles (2013). They showed that, in a new Keynesian model, it is possible to replicate a negative nominal interest rate by implementing a rising path of consumption taxes, to induce agents to front-load their demand for consumption, together with a falling path of labor income taxes, to o¤set the resulting distortion to labor supply. In the presence of capital, this policy must be supplemented with a temporary capital subsidy. However, they only considered a temporary liquidity trap. By contrast, under secular stagnation, the economy is permanently liquidity trapped. I therefore restrict my attention to stationary policies. Eggertsson (2010) also showed that some …scal instruments can be relied upon to stimulate aggregate demand.

The paper is organized as follows. Section 2 presents the representative household model of secular stagnation. In Section 3, I discuss some important properties of the secular stagnation equilibrium. In Section 4, I investigate the e¤ectiveness of conventional macroeconomic stabilization policies. The optimal policy is derived in Section 5. The implications of heterogeneity are brie ‡y discussed in Section 6. The paper ends with a conclusion.

The Representative Household Model 2.1 Households

Time is continuous. There is a mass 1 of in…nitely lived households. Each household consumes a quantity c t of a single consumption good with price P t . It supplies labor l t paid at nominal wage W t and receives a real lump-sum transfer t . The nominal wealth A t of a household is composed of physical capital K t , government bonds B t , and money M t :

A t = P K t K t + B t + M t ; (1)
where P K t denotes the nominal price of capital. Firms pay R t to rent capital from households. Capital depreciates at rate . There are three components to the nominal return from a unit of physical capital: the rent R t , the loss due to depreciation P K t , and the capital gain _ P K t . Bonds yield a nominal return i t , while money yields a zero nominal return. The wealth of the representative household therefore evolves according to:

_ A t = h R t P K t + _ P K t i K t + i t B t + W t l t + P t t P t c t ; = i t A t + " R t P K t + _ P K t P K t i t # P K t K t i t M t + W t l t + P t t P t c t : (2)
Let t = _ P t =P t denote the rate of in ‡ation, r t = i t t the real interest rate, w t = W t =P t the real wage, a t = A t =P t the real household wealth, m t = M t =P t the real money holdings, and p K t = P K t =P t the real price of capital, measured in units of consumption good. Dividing the wealth accumulation equation by the price level P t , while using the fact that _ A t =P t = _ a t + t a t and _

P K t =P K t = _ p K t =p K t + t , yields: _ a t = r t a t + R t P K t + _ p K t p K t r t p K t K t i t m t + w t l t + t c t : (3)
In the absence of uncertainty, by arbitrage, the returns from holding capital must be equal to the returns from holding bonds. Thus, the second term on the right hand side of (3) must be equal to zero, which yields the following relationship between the user cost of capital and the real interest rate:

r t = R t P K t + _ p K t p K t : (4)
The wealth accumulation equation therefore simpli…es to:

_ a t = r t a t i t m t + w t l t + t c t : (5)
The intertemporal budget constraint prevents households from running Ponzi schemes:

lim t!1 e R t 0 rsds a t 0: (6)
The representative household discounts the future at rate > 0. At any point in time, it derives utility u (c t ) from consuming c t , with u 0 ( ) > 0, u 00 ( ) < 0, and lim c!0 u 0 (c) = 1. It also derives utility h (m t ) from holding real money balances m t , with h 0 ( ) > 0, h 00 ( ) < 0, lim m!0 h 0 (m) = 1, and h 0 (m) = 0 for all m m. At m, the household is satiated with real money balances and does not derive any utility from holding more money for transaction purposes. The household incurs disutility v (l t ) from supplying labor l t , with v 0 ( ) > 0, v 00 ( ) > 0, v 0 (0) = 0, and lim l! l v 0 (l) = 1 where l is the maximum feasible supply of labor, which can be in…nite. The household also derives utility from holding wealth a t . However, it knows that it will eventually need to cover the liabilities of the government, which are composed of the real aggregate supply of public debt b s t and of money m s t . The utility derived by a household from holding wealth is therefore equal to (a t (b s t + m s t )), with 0 ( ) > 0, 00 ( ) 0, and lim K!0 0 (K) < 1.

Finally, at any time t, utility is additively separable between its four components. The intertemporal utility function is therefore given by:

Z 1 0 e t [u (c t ) v (l t ) + h (m t ) + (a t (b s t + m s t ))] dt: (7)
Including the government liabilities b s t + m s t into the utility function ensures that the Ricardian equivalence holds and that the model has reasonable welfare properties. Indeed, if households did derive utility from holding wealth independently of the level of government liabilities, then the government would be able to arti…cially increase welfare by giving households a huge lump-sum subsidy that would eventually be o¤set by a huge lump-sum tax. This would increase both the level of public debt b s t and the wealth a t of households and, hence, their welfare. At the zero lower bound, money and bonds are perfect substitutes and welfare could also be arti…cially increased by raising the aggregate money supply m s t . I therefore assume that the government cannot mechanically raise welfare by increasing the level of public liabilities. Note that, in the interesting special case where the marginal utility of wealth is constant, the behavior of households is identical whether or not government liabilities are included in the utility function.

The household's problem is to maximize intertemporal utility, (7), subject to the budget constraint, ( 5) and ( 6), with a 0 given. Importantly, the paths of public debt b s t and of aggregate money supply m s t from time 0 onwards are exogenous to the representative household's actions. By the maximum principle, the solution to the household's problem is characterized by:

_ c t c t = r t + 0 (a t (b s t + m s t )) u 0 (c t ) u 0 (c t ) u 00 (c t ) c t ; (8) v 0 (l t ) = w t u 0 (c t ) ; (9)
h 0 (m t ) = i t u 0 (c t ) ; (10)
together with the transversality condition:

lim t!1 e t u 0 (c t ) a t = 0: (11)
When i t = 0, there is no opportunity cost of holding money rather than bonds. In that case, the household's utility from real money balances must be satiated. Moreover, when i t = 0, money and bonds are both zero interest yielding assets that can be used for saving, i.e. they are perfect substitutes. The household is therefore happy to rely on money for saving and, hence, its demand for real money balances can be anything greater or equal to m. The zero lower bound on the nominal interest rate i t therefore follows from the money demand equation (10).

Firms

Firms rent capital K d t from households and employ labor L t to produce output y t using a constant returns to scale neoclassical production function:

y t = F K d t ; L t : (12)
They choose their demand for capital K d t and for labor L t such as to maximize their pro…ts:

P t F K d t ; L t R t K d t W t L t : (13)
In equilibrium, each factor of production must be paid its marginal product:

R t P t = F K K d t ; L t ; (14)
w t = W t P t = F L K d t ; L t : (15)
By assumption, one unit of output can either be transformed into a consumption good or an investment good. [START_REF] Buiter | The Simple Analytics of Helicopter Money: Why It Works -Always[END_REF] Hence, if investment is strictly positive, then the price of capital must be equal to that of consumption, i.e. p K t = P K t =P t = 1. In that case, by ( 4) and ( 14), the real interest rate must be given by:

r t = F K K d t ; L t : (16)
This expression applies in steady state where investment must be strictly positive to compensate the depreciation of capital.

Government

Let M s t and m s t denote the nominal and real money supply at time t, respectively. The growth rate of the nominal money supply at t is given by t = _ M s t =M s t . We therefore have M s t = M 0 e R t 0 s ds , where the initial money supply M 0 is assumed to be exogenously given. [START_REF] Caballero | The Safety Trap[END_REF] The supply of money by the government is therefore characterized by t for all t 2 [0; +1).

At any time t, the government gets real revenue _ M s t =P t from seigniorage, it incurs real expenditures g t , and it makes a lump-sum transfer t to each household. Thus, the nominal supply B s t of government bonds evolves according to:

_ B s t = i t B s t + P t t + P t g t _ M s t : (17)
Note that t m s t = _ M s t =P t = _ m s t + t m s t . Thus, expression (18) for the accumulation of public debt can also be written as:

_ b s t + _ m s t = r t [b s t + m s t ] + t + g t i t m s t : (19)
Integrating the debt accumulation equation (18) gives:

e R t 0 rxdx b s t = b 0 + Z t 0 e R s
where b 0 = B 0 =P 0 with B 0 exogenously given. Integrating instead equation (19) yields:

e R t 0 rxdx (b s t + m s t ) = b 0 + m 0 + Z t 0 e R s 0 rxdx [ s + g s i s m s s ] ds: (21)
In this setup, we can consider two versions of the intertemporal government budget constraint. The …rst is given by the following no-Ponzi condition:

lim t!1 e R t 0 rxdx (b s t + m s t ) 0: (22)
This implies that money is a liability to the government, which must therefore asymptotically have enough wealth to be able to redeem the outstanding money supply. This is the most conservative version of the intertemporal budget constraint. It corresponds to a government that behaves responsibly and that always wants to be in full control of the money supply. Hence, seigniorage cannot be a permanent source of revenue. Indeed, as can be seen from ( 21), by issuing money rather than bonds, the government only economizes on the nominal interest rate. This no-Ponzi condition (22) is the most common form of the intertemporal government budget constraint found in the literature (see, for instance, Kocherlakota and Phelan 1999).

The lax version of the intertemporal government budget constraint is given by:

lim t!1 e R t 0 rxdx b s t 0: (23)
Thus, asymptotically, the present value of debt cannot be positive. However, this does not put any restriction on the ability of the government to print money in order to pay down its debts. Indeed, equation (20) shows that, under the no-Ponzi condition (23), seigniorage is a source of public revenue. This implies that a government that issues debt in its own currency cannot be insolvent. Buiter (2014) forcefully argues that this is the intertemporal budget constraint that governments legally face as they do not have any obligation to redeem the outstanding money supply.

For most of my analysis, I will consider the conservative version of the government budget constraint, given by (22). Indeed, a government with a strong commitment to keeping in ‡ation low wants to be able to redeem its outstanding money supply under all circumstances. Moreover, I want to characterize policies that can lift the economy out of depression without resulting in a huge accumulation of money or public debt that would be very dangerous should economic circumstances unexpectedly change in the future (as would be the case if households suddenly ceased to have a preference for wealth). I will nevertheless occasionally comment on the consequences of the lax version of the government budget constraint.

Market Clearing

In a frictionless equilibrium, …ve market clearing conditions must be satis…ed. In the market for goods, production must be equal to sum of consumption, investment, and government expenditures:

y t = c t + K t + _ K t + g t ; ( 24
)
where investment is equal to K t + _ K t . In the labor market, …rms'labor demand L t must be equal to workers'labor supply l t :

L t = l t : (25)
The market for physical capital clears when the quantity K t supplied by households through their savings is equal to the quantity K d t demanded by …rms:

K t = K d t : (26)
In the bond market, households'demand for bonds B t must be equal to the government supply B s t :

B t = B s t : (27)
Finally, in equilibrium, the market for money must also clear. The household's nominal demand for money is equal to P t m t , while the supply of money by the government is equal to M s t = M 0 e R t 0 s ds . Thus, the market for money is in equilibrium provided that:

m t = e R t 0 ( s s)ds M 0 =P 0 ; (28)
where I have used the fact that P t = P 0 e R t 0 sds .

Frictions

If prices, interest rates, and wages were unconstrained, then the …ve market clearing conditions would be satis…ed. However, to obtain a stationary secular stagnation equilibrium, where the economy fails to produce at full capacity, I impose some restrictions on the adjustment of interest rates and of wages.

In an economy with …at currency, a fundamental constraint on the equilibrium is that the nominal interest rate cannot be negative:

i t 0: (29)
Indeed, no-one is willing to pay more than 100$ for a bond yielding 100$ in a year. People would rather choose to rely on money for saving. This follows from the money demand equation (10).

Prices are perfectly ‡exible. The central bank never allows in ‡ation to rise above a ceiling . Thus, in equilibrium, we must have:

t : (30)
The combination of the zero lower bound on the nominal interest rate and of the upper bound on in ‡ation results in a lower bound on the real interest rate:

r t : (31)
Note that nothing in the model justi…es the existence of an upper bound on in ‡ation. However, it is a key ingredient of secular stagnation that does accurately describe the behavior of all major central banks around the world. If wages were perfectly ‡exible, then they would be given by the marginal product of labor (15) with labor demand L t equal to labor supply l t :

w t = F L (K t ; l t ) : (32)
However, a binding zero lower bound on the nominal interest rate could result in a de ‡ationary feedback loop, whereby a lack of demand induces a fall in prices, which raises the real interest rate, which further reduces demand. This is inconsistent with the existence of a stationary equilibrium. Hence, to put a break on the extent of de ‡ation, I assume that wages are downward rigid.

Recall that the real wage w t must always be equal to the marginal product of labor F L (K t ; L t ), as otherwise …rms can increase pro…ts by adjusting employment. For …xed values of the stock of physical capital K t and of labor demand L t , the nominal wage

W t = P t F L (K t ; L t )
increases by the rate of in ‡ation t . I impose the following lower bound on wage growth for unchanged values of K t and L t :

(1 + t dt) W t 1 + R dt W t ( dt) [W t P t F L (K t ; l t )] ; (33)
where R denotes the reference rate of in ‡ation used in the wage bargaining process and 0 denotes the speed of convergence of the wage to the marginal product of labor at full employment.

I assume that W t P t F L (K t ; l t ) and the wage rigidity constraint (33) must hold with complementary slackness. Thus, if W t > P t F L (K t ; l t ), then it must be due to the binding wage rigidity constraint. Workers would like their nominal wage to increase at rate R , but are willing to accept a smaller increase to reduce the gap between their current wage W t and the marginal product of labor at full employment P t F L (K t ; l t ). Conversely, if the wage rigidity constraint is not binding, then the wage W t must be equal to the marginal product of labor at full employment P t F L (K t ; l t ).

The lower bound on wage growth is speci…ed for …xed values of K t and L t , i.e. the left hand side of the inequality is (1 + t dt) W t = (1 + t dt) P t F L (K t ; L t ) = P t+dt F L (K t ; L t ) rather than W t+dt = P t+dt F L (K t+dt ; L t+dt ). I am therefore assuming that, in the wage bargaining process, workers are willing to adjust their wage rate if it re ‡ects a change in the stock of physical capital or in the amount of labor employed by …rms. Thus, workers accept a wage cut if …rms employ them for more hours, which reduces the marginal product of labor. By contrast, without this adjustment, the fall in the marginal product of labor following an increase in L t would need to be compensated by higher in ‡ation for the nominal wage P t F L (K t ; L t ) to remain unchanged. This would greatly complicate the out-of-steady-state dynamics of in ‡ation. [START_REF] Carroll | Why Do the Rich Save So Much?[END_REF] Note that, in steady state, both K t and L t are constant and these considerations are therefore irrelevant.

Two natural benchmarks are R = 0, which corresponds to a reluctance by workers to accept nominal wage cuts, and R = , in which case workers would like their nominal wage to increase by the in ‡ation ceiling, which might also be the central bank's in ‡ation target. [START_REF] Correia | Unconventional Fiscal Policy at the Zero Bound[END_REF] Throughout my analysis, I shall consider that: R :

(34) Thus, workers never bargain for nominal wages that increase faster than the in ‡ation ceiling. If = 0, then nominal wages for given values of K t and L t cannot increase by less than the reference rate of in ‡ation R , independently of the amount l t of labor supplied; while, as ! 1, wages become perfectly ‡exible.

The wage rigidity constraint (33) can be written in real terms as:

t R w t F L (K t ; l t ) : (35)
In equilibrium, we must have t 2 R ; +1 as, otherwise, the constraint cannot be satis…ed. The downward wage rigidity assumption is that (35) and w t F L (K t ; l t ) must hold with complementary slackness. Thus, if the wage rigidity constraint is slack, then

w t = F L (K t ; l t ), which, by (35), implies t 2 R ; +1 . Hence, if t 2 R
; R , then the wage rigidity constraint must hold with equality. If w t > F L (K t ; l t ), then the wage rigidity constraint must be binding, which implies that t 2 R ; R . Hence, if t 2 R ; +1 , then w t = F L (K t ; l t ). The real wage must therefore satisfy:

w t = ( F L (K t ; l t ) t ( R ) F L (K t ; l t ) if t 2 R ; +1 if t 2 R ; R (36)
Recall that we always have w t = F L (K t ; L t ). Hence, if in ‡ation is above the reference rate R of in ‡ation, then the downward wage rigidity is not binding and the labor market clears, i.e. L t = l t . If, however, in ‡ation is below the reference rate R , then the downward wage rigidity maintains the real wage above the marginal product of labor at full employment, which generates a labor demand L t that falls short of the labor supply l t . For a given rate t of in ‡ation smaller than R , a rise in wage ‡exibility as measured by reduces the discrepancy between labor demand and labor supply.

Equilibrium

An equilibrium is de…ned as follows.

De…nition

, K 0 , b 0 = B 0 =P 0 , and m 0 = M 0 =P 0 ; y t ; L t ; K d t 1
t=0 solves the producer's problem given P 0 and ( t ; w t ; R t ) 1 t=0 ;

( t ; g t ; t ; b s t ) 1 t=0 satis…es the intertemporal government budget constraint given (r t ; i t ; t ) 1 t=0 , b 0 = B 0 =P 0 , and m 0 = M 0 =P 0 ; Equilibrium prices P 0 and r t ; i t ; t ; w t ; R t ; p K t 1 t=0 are such that markets clear:

-The market for goods clears:

y t = c t + K t + _ K t + g t ;
-The market for physical capital clears:

K t = K d t ; -The bond market clears: b t = b s t ; -The market for money clears: m t = e R t 0 ( s s)ds M 0 =P 0 ;
-The real wage is given by (36). [START_REF] Eggertsson | What Fiscal Policy is E¤ective at Zero Interest Rates?[END_REF] I …rst characterize the steady state equilibria of this economy. I then discuss their stability. Throughout this section, I set the government expenditures to zero, i.e. g t = 0, and impose the conservative version of the intertemporal government budget constraint, given by (22). I take the growth rate t of the money supply as given and subsequently characterize the conditions that it needs to satisfy in equilibrium. I assume that the lump-sum transfer t is constant over time and that it adjusts such as to have a binding intertemporal government budget constraint given (i t ; r t ; t ) 1 t=0 , b 0 = B 0 =P 0 , and m 0 = M 0 =P 0 . 9 The equilibrium level of government debt b t is determined by the resulting policy, according to (20).

Steady State Equilibria

In a steady state equilibrium, consumption c t , labor supply l t , labor demand L t , physical capital K t , the real wage w t , and the real interest rate r t must all be constant over time. Without loss of generality, we can also consider that both in ‡ation t and the nominal interest rate i t are constant over time. [START_REF] Eggertsson | The Zero Bound on Interest Rates and Optimal Monetary Policy[END_REF] I subsequently drop the time subscripts from those variables to refer to their steady state values.

For the capital stock to be constant, investment needs to be strictly positive to o¤set depreciation. In steady state, the real price of capital p K t must therefore be equal to 1. Hence, the real interest rate is just equal to the marginal product of capital net of depreciation, cf. (16). Note that, by (1), when p K t = 1, wealth net of government liabilities a t (b t + m t ) must be equal to the stock of physical capital K.

The simplest way to characterize the steady state equilibria of the economy is to look at it through the lens of the Aggregate Demand -Aggregate Supply paradigm. Fundamentally, aggregate demand is given by y = c + K, while aggregate supply is given by y = F (K; L). I therefore need compute these levels of demand and supply for each rate of in ‡ation. For this, I will rely on the characterization of (c; l; L; K; w) as a function of the real interest rate r, which is jointly obtained from the optimality conditions for consumption (8) with _ c t = 0 and a t (b t + m t ) = K and for labor supply (9), from the demand for labor (15) and for capital (16), and from the expression for the real wage (36).

Clearly, if in ‡ation is high enough, then the zero lower bound on the nominal interest rate is not binding and the real interest rate can be su¢ ciently small to induce the economy to produce at full capacity. Let denote the smallest rate of in ‡ation such [START_REF] Eggertsson | A Model of Secular Stagnation[END_REF] Recall that m s t is equal to e R t 0 ( s s )ds M 0 =P 0 , which is why t and t do enter the intertemporal government budget constraint. [START_REF] Eggertsson | The Zero Bound on Interest Rates and Optimal Monetary Policy[END_REF] If i t = 0, then r t = t . If t < R , then in ‡ation determines the gap between w t and F L (K t ; l t ), through (36). Thus, in these two cases, steady state in ‡ation must be constant. And, if i t > 0 and t R , then money is super-neutral. Hence, there is no loss of generality in considering that in ‡ation is constant over time.

that the zero lower bound is not binding. I assume that > , which implies that the in ‡ation ceiling is not high enough to allow the economy to produce at full capacity. Recall that, by (34), the reference rate of in ‡ation R is assumed to be lower than the in ‡ation ceiling . We therefore have:

> R . ( 37
)
By ( 15) and (36), the labor market clears, i.e. L = l, when 2 R ; +1 , but fails to clear, i.e. L < l, when 2 R ; R . We can therefore distinguish three intervals of interest. When 2 [ ; +1), the frictions are irrelevant. When 2 R ; , the zero lower bound is binding but the labor market clears. Finally, when 2 R

; R , the zero lower bound is binding and the labor market does not clear. [START_REF] Friedman | The Optimum Quantity of Money[END_REF] Thus, when 2 [ ; +1), the economy must be in a frictionless equilibrium.

Lemma 1 A steady state frictionless equilibrium always exists and, if is su¢ ciently close to zero and labor supply is su¢ ciently inelastic, then it must be unique.

Note that, even if is high and labor supply is highly elastic, it is di¢ cult to …nd a counter-example with multiple equilibria. I therefore consider throughout my analysis that the frictionless equilibrium is unique. Let r n denote the natural real interest rate, i.e. the real interest rate of the frictionless equilibrium. Recall that (c; l; L; K; w) can be characterized as a function of the real interest rate. The natural rate r n is therefore determined such as to equate aggregate demand y = c + K to aggregate supply y = F (K; L). The corresponding nominal interest rate is given by i = r n + . By de…nition, is the smallest rate of in ‡ation at which frictions are irrelevant. Hence, when in ‡ation is equal to , we must have i = 0. This implies that = r n . [START_REF] Hansen | Economic Progress and Declining Population Growth[END_REF] When > , the frictionless equilibrium is inconsistent with the in ‡ation ceiling . The condition that > is fundamentally an assumption on the natural real interest rate, which must satisfy r n = < . In particular, if 0, the natural rate must be negative. The optimality condition for consumption (8) implies that, in steady state, r = 0 (K) =u 0 (c). Hence, if is su¢ ciently small, the real interest rate is indeed likely to be negative. This shows that the preference for wealth decreases the natural real interest rate to such an extent that it can easily make the frictionless equilibrium inconsistent with the in ‡ation ceiling . [START_REF] Friedman | The Optimum Quantity of Money[END_REF] Recall that, by the wage rigidity constraint (35), R is inconsistent with the existence of an equilibrium.

12 It can easily be shown that < . In equilibrium, the marginal product of capital F K (K; L) must be strictly positive. Hence, by ( 16), we must have r n + = F K (K; L) > 0, which implies that = r n < . When 2 R ; , the labor market clears but the real interest is excessively high, as r = i > 0 = r n . Under mild conditions, this is inconsistent with the existence of a steady state equilibrium.

Lemma 2 Aggregate demand y = c + K is strictly smaller than aggregate supply y = F (K; L) for all 2 R ; provided that is su¢ ciently close to zero and either that labor supply is su¢ ciently inelastic or that c > K for all r 2 [r n ; ]. [START_REF] Keynes | The General Theory of Employment, Interest and Money[END_REF] Note that, if consumption c is greater than investment K and if > , then c > K.

We can now prove the existence and uniqueness of a steady state equilibrium for 2 R ; R provided that wages are su¢ ciently sticky, i.e. provided that is su¢ ciently small. I consider that the growth rate of the money supply is su¢ ciently strong to have a binding zero lower bound in steady state equilibrium. [START_REF] Kocherlakota | Impact of Land Price Fall When Labor Markets are Incomplete[END_REF] Indeed, aggregate demand would be even more depressed with a strictly positive nominal interest rate.

Lemma 3 If aggregate demand is strictly smaller than aggregate supply when = R (as implied by Lemma 2) and if < R + , then there exists at least one steady state equilibrium such that 2 R ; R . If is su¢ ciently close to zero, this must be the unique steady state equilibrium with i = 0 and < . I henceforth consider that the secular stagnation equilibrium is unique. When 2

R

; R , the wage rigidity constraint is binding, which implies that …rms' labor demand L is smaller than households' labor supply l. This clearly shows that, in the secular stagnation equilibrium, the economy fails to produce at full capacity.

Figure 1 displays the aggregate demand and aggregate supply curves, assuming that i = 0 and r = for all 2 R ; . When 2 [ ; +1), the zero lower bound is not binding. The economy therefore is in the frictionless equilibrium where the real interest rate adjusts such as to equate aggregate demand to aggregate supply. Hence, the two curves overlap at the natural level of output for all 2 [ ; +1).

By Lemma 2, we know that aggregate supply is greater than aggregate demand whenever 2 R ; . Note that, as the real interest rate increases from to R , aggregate supply can either rise or fall. On the one hand, an increase in the real rate reduces K=L, by (16), which contracts aggregate supply y = F (K; L) = LF (K=L; 1). On the other [START_REF] Keynes | The General Theory of Employment, Interest and Money[END_REF] The consumption Euler equation (8) implies that in steady state r , which is why we can ignore the possibility that r > . [START_REF] Kocherlakota | Impact of Land Price Fall When Labor Markets are Incomplete[END_REF] A su¢ cient condition is 2 R ; . Indeed, to have a steady state equilibrium with i > 0, in ‡ation must be equal to the money growth rate, by (10) and (28). Hence, if = 2 R ; and i > 0, then r = i > > = r n . But, by Lemma 2, there is no steady state equilibrium with r > r n and 2 R ; . Hence, when 2 R ; , there is no steady state equilibrium with i > 0.

hand, a higher real rate reduces the demand for consumption, by (8), which can raise the supply of labor, by (9), which expands aggregate supply, since L = l when R . [START_REF] Kocherlakota | Explaining the Fiscal Theory of the Price Level[END_REF] Finally, assuming that wages are su¢ ciently sticky, we know by Lemma 3 that an equilibrium always exists for some 2 R ; R . Indeed, as in ‡ation falls from R to R , labor demand L shrinks from l to 0, by (36). The capital stock K also falls in order to maintain a capital-labor ratio K=L consistent with the real interest rate r = , by ( 16). This generates a much stronger contraction in aggregate supply than in aggregate demand, as the fall in K and L does not a¤ect the consumption component of aggregate demand. ; R . The economy is therefore confronted with persistent under-employment, as L < l, and with an in ‡ation rate that is persistently below the ceiling.

Let us now characterize the monetary policy that is consistent with the di¤erent equilibria of the economy. In a frictionless equilibrium with 2 ( ; +1), the nominal interest rate is constant and strictly positive, as i = r n + > r n + = 0. The money demand equation (10) therefore implies that real money balances m t must also be constant over time. But, by the money market clearing condition (28), this requires the growth rate t of the money supply to be equal to the in ‡ation rate . We must therefore have = 2 ( ; +1) The initial price level P 0 is then trivially determined by:

P 0 = M 0 m , ( 38
)
where m is given by the money demand equation ( 10) with i = r n + , while M 0 is exogenously given. [START_REF] Kopczuk | Bequest and Tax Planning: Evidence from Estate Tax Returns[END_REF] In the secular stagnation equilibrium, the zero lower bound is binding. Thus, the money demand equation (10) implies that m t m. The initial price level must therefore satisfy M 0 =P 0 m or, equivalently, P 0 M 0 = m. Indeed, when i = 0, money and bonds are perfect substitutes. Hence, while the transaction demand for money never rises above m, the extra money supplied m t m 0 can be used for savings. Assuming that, initially, households exclusively rely on bonds for savings, the price level must be given by:

P 0 = M 0 m . ( 39
)
In that case, to have an equilibrium with i = 0, the real supply of money e

R t 0 ( s )ds M 0 =P 0 = e R t 0 ( s
)ds m must always be greater or equal to m. Hence, such an equilibrium with a given rate 2 R ; R of in ‡ation is consistent with any path of the growth rate of the money supply that satis…es R t 0 ( s ) ds 0 for all t. Thus, if the central bank adopts a constant growth rate of the money supply, then a liquidity trap equilibrium with in ‡ation 2 R ; R requires . The last remaining possibility is to have a frictionless equilibrium with in ‡ation exactly equal to . In that case, the nominal interest rate is equal to zero. Hence, in theory, a rise in the money supply is not necessarily in ‡ationary. However, if a strictly positive mass of households decides to spend the extra money supplied, rather than save it, then in ‡ation and the nominal interest rate both rise. Thus, the liquidity trap equilibrium with in ‡ation equal to is not stable. This is fundamentally due to the fact that, although [START_REF] Kopczuk | Bequest and Tax Planning: Evidence from Estate Tax Returns[END_REF] It is well known (cf. Obstfeld and Rogo¤ 1983) that several paths of the price level are consistent with a given path of the money supply. I choose to focus on the path of the price level that results in a constant rate of in ‡ation. This equilibrium selection is consistent with the quantity theory of money as a once-and-for-all increase in the money supply results in a once-and-for-all increase in the price level of the same magnitude (cf. Kocherlakota and Phelan 1999). But, by ( 10), (28), and i t = r n + t , any path of t and t that satisfy h 0 e R t 0 ( s s )ds M 0 =P 0 = [r n + t ] u 0 (c) with t for all t is consistent with the steady state frictionless equilibrium.

the nominal interest rate is equal to zero, the zero lower bound is not binding as it is not preventing the nominal rate from being negative. We can therefore consider that, in a frictionless equilibrium with i = 0 and = , the real quantity of money must be exactly equal to m and the growth rate of the money supply must be exactly equal to . In that case, the initial price level P 0 is equal to M 0 = m.

Finally, we need to check that the representative household's intertemporal budget constraint (6) and transversality condition (11) are both satis…ed. Recall that the lumpsum transfer adjusts such as to balance the intertemporal government budget constraint. It follows that the government's no-Ponzi condition (22) holds with equality. But, in steady state, the real interest rate is constant and the wealth of the representative household is given by a t = K + b t + m t . It immediately follows that:

lim t!1 e R t 0 rsds a t = lim t!1 e rt K + lim t!1 e rt (b t + m t ) ; = lim t!1 e rt K 0: (40)
The household's intertemporal budget constraint is therefore binding when r > 0, but not when r 0.

The optimality condition for consumption (8) implies that, in steady state, r. It follows that:

lim t!1 e t u 0 (c t ) a t = lim t!1 e t u 0 (c) (K + b t + m t ) ; = lim t!1 e t u 0 (c) K + lim t!1 e ( r)t u 0 (c) e rt (b t + m t ) ; = 0 + lim t!1 e ( r)t u 0 (c) lim t!1 e rt (b t + m t ) ; = 0: (41)
Thus, the household's transversality condition is always satis…ed, even when its budget constraint is not binding. This con…rms the optimality of the behavior of the representative household.

Stability of Equilibrium

We have shown that there are essentially two steady state equilibria: a frictionless steady state that requires in ‡ation to be greater or equal to and a secular stagnation steady state with in ‡ation smaller than R . I now investigate the stability properties of these equilibria.

In a world with no lower bound on the real interest rate and no downward wage rigidity, any steady state equilibrium must be frictionless. If there is a unique frictionless steady state and if labor supply is inelastic, then a phase diagram reveals that the frictionless equilibrium must be globally saddle-path stable. By continuity, this must also be true if labor supply is su¢ ciently inelastic. [START_REF] Kopczuk | Economics of Estate Taxation: A Brief Review of Theory and Evidence[END_REF] Let us now turn to the stability of the secular stagnation steady state, which is the only feasible steady state equilibrium under the in ‡ation ceiling . [START_REF] Krugman | It's baaack: Japan's Slump and the Return of the Liquidity Trap[END_REF] If the zero lower bound and the downward wage rigidity are both binding and if the real price p K t of capital is equal to 1, then the secular stagnation steady state is locally unstable provided that is close to zero, i.e. that wages are very rigid, and that in steady state [START_REF] Kumhof | Inequality, Leverage, and Crises[END_REF] This can easily be seen in the special case where is equal to 0, which implies that in ‡ation must be equal to R , and where the marginal utility of wealth is constant, i.e. 00 (K) = 0. Let us assume for a contradiction that the economy converges to its secular stagnation steady state. With constant in ‡ation and constant marginal utility of wealth, the consumption Euler equation ( 8) implies that consumption immediately jumps to its steady state value c. Similarly, the demand for capital (16) implies that the capital-labor ratio K t =L t also immediately jumps to its steady state value K=L. The capital accumulation equation could then be written as:

K 00 (K) = 0 (K) < cu 00 (c) =u 0 (c).
_ K t = F (K t ; L t ) K t c t ; = K t F 1; 1 K=L c: (42)
Clearly, if K t is above its steady state value K, then K t must diverge o¤ to in…nity; while if K t is below K, then it must diverge towards zero. However, physical capital is the only state variable of the economy. Also, if there is an upper bound to labor supply, then physical capital must belong to a closed interval. Hence, if an equilibrium exists, then physical capital must be converging towards a steady state. [START_REF] Michaillat | An Economical Business-Cycle Model[END_REF] It follows that in the neighborhood of the secular stagnation steady state, we must either have a non-binding wage rigidity constraint or p K t < 1. 21 Let us …rst consider the scenario where the economy is in the frictionless equilibrium with in ‡ation and where the central bank unexpectedly lowers the in ‡ation ceiling to . The real interest rate immediately increases to , which is higher than the marginal [START_REF] Kopczuk | Economics of Estate Taxation: A Brief Review of Theory and Evidence[END_REF] It can also be shown, by linearizing (8) with r t = F K (K t ; l t ) and a t (b t + m t ) = K t and (24) with y t = F (K t ; l t ) around the steady state (c; K) while using (9) with w t = F L (K t ; l t ) to express l t as a function of c t and K t that, if a steady state satis…es r n = F K (K; l) 0 and c > K, then it must be locally saddle-path stable. [START_REF] Krugman | It's baaack: Japan's Slump and the Return of the Liquidity Trap[END_REF] I am now implicitly assuming that the growth rate of the money supply is su¢ ciently high to avoid having a steady state with a positive nominal interest rate and < . By footnote 14, a su¢ cient condition for this is 2 R ; . [START_REF] Kumhof | Inequality, Leverage, and Crises[END_REF] This can be shown by linearizing the system around the secular stagnation steady state. [START_REF] Michaillat | An Economical Business-Cycle Model[END_REF] As in the standard Ramsey model, K t = 0 is not a stable steady state equilibrium. [START_REF] Obstfeld | Speculative Hyperin ‡ations in Maximizing Models: Can We Rule Them Out?[END_REF] Note that, as the zero lower bound is binding in the secular stagnation steady state, we cannot have a strictly positive nominal rate while the economy is asymptotically converging to that steady state. Hence, we cannot relax the assumption that i t = 0, while considering that the wage rigidity constraint is binding and that p K t = 1.

product of capital net of depreciation. But, by arbitrage, the returns from holding bonds must always be equal to the returns from holding physical capital, cf. ( 4) and ( 14). The price p K t of capital must therefore fall below 1 such as to satisfy the following relationship between the real interest rate t and the marginal product of capital:

t = F K (K t ; L t ) p K t + _ p K t p K t : (43)
Aggregate investment drops to zero. The capital stock therefore falls are rate , since

_ K t = K t .
Both the fall in the capital stock and the higher real interest rate reduce consumption. This results in a sharp contraction in aggregate demand and, therefore, in the demand for labor, which in the absence of investment is implicitly determined by

F (K t ; L t ) = c t .
The fall in labor demand below labor supply induces in ‡ation to drop below R .

The capital stock falls until it reaches the secular stagnation steady state. At this point, the price of capital becomes equal to 1, and investment becomes positive. This induces a discrete rise in labor demand, which raises the marginal product of capital and equates it to the steady state real interest rate.

If the initial stock of capital is below the secular stagnation steady state, then it must be rising. This requires positive investment and, hence, a price of capital p K t equal to 1. To avoid diverging away from the steady state, the downward wage rigidity must not be binding. Hence, the economy must be operating at full capacity, with L = l, while converging towards secular stagnation. The dynamics of the frictionless economy are at work, except that the economy is not on a path converging to the frictionless steady state. Once the capital stock reaches its secular stagnation value, the economy suddenly stalls. Indeed, investors know that any further increase in the capital stock will eventually result in the price of capital dropping below 1 and, hence, in a capital loss.

Once in secular stagnation, investment drops as _ K t , which was strictly positive, suddenly becomes equal to zero. This generates a fall in labor demand, which becomes smaller than labor supply. This raises the capital-labor ratio. The real interest rate therefore falls, such that in steady state r = 2 R ; R . Importantly, this implies that the real interest rate just before falling into secular stagnation was higher than R and, hence, higher than the lower bound . Interestingly, both the convergence from above or from below imply that the secular stagnation steady state is reached in …nite time.

Properties of the Secular Stagnation Equilibrium

If the central bank does not allow in ‡ation to rise above , with < = r n , then the economy must eventually fall into the secular stagnation steady state. Before investigating its properties, I begin by deriving a useful comparative statics result. We expect an exogenous rise in the real interest rate to reduce both consumption and output. [START_REF] Piketty | On the Long-Run Evolution of Inheritance: France 1820-2050[END_REF] The following lemma provides a necessary and su¢ cient condition for this.

Lemma 4 In the secular stagnation steady state, an exogenous rise in the real interest rate reduces both consumption and output if and only if K 00 (K) = 0 (K) < cu 00 (c) =u 0 (c).

Throughout my analysis, I therefore assume that, in the secular stagnation steady state, K 00 (K) = 0 (K) < cu 00 (c) =u 0 (c). This condition must be satis…ed for all the propositions of this section to hold. It does indeed seem natural to consider that the utility of wealth is less concave than the utility of consumption.

Under secular stagnation, the economy is permanently depressed with some unemployed labor resources. However, the fact that the labor demand of …rms is persistently below the labor supply of workers is not fundamentally due to the wage rigidity. Indeed, when wages are more ‡exible, the economy is more depressed. This is known as the paradox of ‡exibility, which is formalized in the following proposition.

Proposition 1 In the secular stagnation equilibrium, if is su¢ ciently close to zero, then a rise in , i.e. a rise in wage ‡exibility, reduces in ‡ation, which lowers the output level.

The intuition for the result is straightforward. If wages are more ‡exible, then de ‡ation is stronger. This raises the real interest rate, which reduces the demand for consumption and for investment, resulting in a lower output level. Note that the su¢ cient condition of the proposition, that is su¢ ciently close to zero, is far from necessary. [START_REF] Summers | Re ‡ections on the 'New Secular Stagnation Hypothesis[END_REF] The downward wage rigidity breaks the de ‡ationary spiral and is therefore necessary to have a steady state equilibrium, as shown by Lemma 3. However, it is clearly not [START_REF] Piketty | On the Long-Run Evolution of Inheritance: France 1820-2050[END_REF] The exogenous rise in the steady state real interest rate can be due to a reduction in the growth rate of the money supply that raises the nominal rate. If is su¢ ciently close to zero, i.e. if the interval R ; R is su¢ ciently small, then the rise in the nominal interest rate i must also unambiguously raise the real interest rate i , since 2 R ; R . [START_REF] Summers | Re ‡ections on the 'New Secular Stagnation Hypothesis[END_REF] To overturn the result with an signi…cantly greater than zero, the elasticity of intertemporal substitution of consumption or the elasticity of labor of labor supply must be so large that a fall in in ‡ation reduces consumption and, hence, labor demand or raises labor supply so much that in ‡ation decreases by an even larger amount. Thus, in equilibrium, a rise in must raise in ‡ation such as to increase labor demand or reduce labor supply. This generates an upward pressure on in ‡ation that is partially o¤set by the downward pressure that is mechanically triggered by the rise in . This mechanism seems extremely implausible. A similar reasoning implies that Proposition 2, 3, and, 4, below, also are much more robust than the su¢ cient condition suggests. the friction at the origin of the economic depression. Secular stagnation is instead due to a lack of demand resulting from the lower bound on the real interest rate. Indeed, the preference for wealth induces households to have such a high propensity to save that, in the absence of frictions, a very low real interest rate is necessary to raise aggregate demand to the level of aggregate supply. In a nutshell, the gap between labor supply and labor demand is not primarily due to the wage being too high, it is instead due to the real interest rate being too high.

The in ‡ation ceiling induces the economy to fall into secular stagnation, where the real interest rate r = 2 R ; R is essentially determined by the downward wage rigidity. At this interest rate, which is actually strictly higher than the lower bound , aggregate demand is very weak. Hence, the labor demand L of …rms falls below the labor supply l of workers to equate aggregate supply to the weak level of aggregate demand. Output is essentially demand determined.

To understand why a lack of demand can reduce the output level of the economy, it is important to realize that a rise in the propensity to save does not necessarily translate into higher investment. In fact, under secular stagnation, a rise in the intensity of the preference for wealth reduces investment. This is known as the paradox of thrift, which is stated in the following proposition.

Proposition 2 In the secular stagnation equilibrium, if is su¢ ciently close to zero, then a rise in the marginal utility of wealth reduces both consumption and investment.

Indeed, a rise in the propensity to save reduces consumption, which contracts the level of aggregate demand, which in turn reduces the labor demand of …rms. In the limit as tends to zero, the real interest rate remains equal to R and the capital-labor ratio must remain constant. This requires a fall in the capital stock and, hence, in steady state investment. If is positive, the reduction in aggregate demand decreases in ‡ation, which ampli…es the contraction in both consumption and investment. Thus, a rise in the marginal utility of wealth lowers the equilibrium level of wealth! By contrast, in the frictionless equilibrium, a rise in the supply of savings, due to an increase in the marginal utility of wealth, reduces the equilibrium real interest rate. This raises the stock of capital. It also raises consumption provided that, initially, the capital stock is not too high. [START_REF] Werning | Managing a Liquidity Trap: Monetary and Fiscal Policy[END_REF] In the absence of frictions, we therefore have the usual result that a rise in the propensity to save raises investment and the capital stock. The fundamental problem under secular stagnation is that the real interest rate is determined by in ‡ation, and hence by the downward wage rigidity, rather than by the supply and demand for loans.

When the economy su¤ers from a lack of demand, expansionary supply shocks can be contractionary. This is the paradox of toil, which is formalized in the following proposition.

Proposition 3 In the secular stagnation equilibrium, if is su¢ ciently close to zero:

A fall in the disutility of labor reduces consumption, investment and, hence, output; Under a Cobb-Douglas aggregate production function, a rise in total factor productivity reduces consumption, investment and, hence, output.

A rise in the supply of labor l, decreases the marginal product of labor at full capacity F L (K; l). By the binding wage rigidity constraint (33), this reduces the growth rate of nominal wages and, hence, the in ‡ation rate. The corresponding rise in the real interest rate r = generates a contraction in the demand for consumption and for investment. The e¤ect of a rise in total factor productivity is slightly more complex. On the one hand, it mechanically raises the marginal product of capital, which induces …rms to raise their capital-labor ratio K=L. On the other hand, thanks to higher productivity and to more capital per worker, the amount of labor L necessary to meet the demand for consumption falls. It turns out that, with a Cobb-Douglas aggregate production function, these two e¤ects exactly cancel out and, hence, total factor productivity has no direct impact on investment K. However, a higher total factor productivity raises the marginal product of labor F L (K; L), which increases the labor supply l of workers. 25 As before, this reduces in ‡ation, which generates a contraction in the demand for consumption and for investment.

In a nutshell, an expansionary supply shock cannot raise output if it fails to generate a corresponding increase in aggregate demand. The problem is that, under secular stagnation, the real interest rate is determined by in ‡ation. By contrast, in the frictionless equilibrium, the real interest rate adjusts such as to equate aggregate demand to aggregate supply. Thus, in the absence of frictions, a rise in the supply of labor induces a fall in the real interest rate such as to generate a rise in the demand for consumption and for investment. 26

Policy Analysis

In the absence of government intervention, the economy is bound to remain permanently depressed. In this section, I therefore investigate the e¤ectiveness of various stimulative policies. As the objective is to permanently escape secular stagnation, I focus on the long-run impact of these policies on the steady state equilibrium of the economy and abstract from transitional dynamics.

The obvious solution to secular stagnation would be to raise the in ‡ation ceiling to at least . However, recent history has shown that, for political and institutional reasons, this is typically not an option, even in crisis struck countries.

When the nominal interest rate is equal to zero, increasing the money supply through open market operations, i.e. by buying bonds, does not have any e¤ect on the economy. Indeed, when i = 0, consumers'transaction demand for real money balances is satiated and money and bonds are perfect substitutes. Hence, any further increase in the money supply is used for savings: the economy is liquidity trapped! An alternative policy option would be to implement a helicopter drop of money, i.e. to give households a …scal transfer …nanced through an increase in the money supply. This policy violates the conservative version of the government's no-Ponzi condition (22). 27 Hence, it is only feasible if the government does not have to be able to redeem the outstanding money supply.

This policy increases both the wealth of households and the level of government liabilities. If households perceive the money supply as a government liability, then the helicopter drop of money does not increase their net wealth a t (b s t + m s t ) and, hence, cannot have any impact on the level of economic activity. If, on the contrary, households do not perceive the money supply as a government liability, then the policy raises their wealth, which reduces their marginal utility of wealth, and increases their demand for consumption, by the Euler equation ( 8) with _ c t = 0. If the transfers are su¢ ciently large to induce the economy to produce at full capacity, then the increase in the money supply must be highly in ‡ationary, which is inconsistent with the low in ‡ation ceiling. Thus, the helicopter drop of money does not seem to be a completely satisfactory solution to secular stagnation. Also, note that, in the special case where the marginal utility of wealth is constant, the policy is always completely ine¤ective.

The remaining conventional stabilization policy is to implement a …scal stimulus. Let us therefore consider that the government relies on lump-sum taxes to …nance a level g of government spending. 28 Aggregate demand becomes equal to c + K + g. Note that a 27 Let (b t ; m t ) 1 t=0 denote the pre-existing paths of real debt and money, which satisfy the conservative no-Ponzi condition lim t!1 e rt (b t + m t ) = 0. The implementation of the helicopter drop of money at time 0 raises the initial real money supply by m0 . Hence, it increases the real government liabilities at time t by mt = m0 P 0 =P t = m0 e t = m0 e rt . We therefore have lim t!1 e rt (b t + m t + mt ) = m0 > 0.

28 If, alternatively, the government spending is …nanced through increases in the money supply, then temporary increase in government expenditures cannot remedy secular stagnation. The increase in government spending must therefore be permanent.

Let " u = cu 00 (c) =u 0 (c) and " = K 00 (K) = 0 (K). In the presence of government spending, an exogenous rise in the real interest rate reduces both consumption and output if and only if (c + g) " u > c" , which is a generalization of Lemma 4. Throughout this section, I assume that this condition is satis…ed.

Proposition 4 In the secular stagnation equilibrium, in the limit as tends to zero, i.e. when wages are completely rigid, the steady state government spending multiplier is equal to:

1 + K + c " "u (c + g) c " "u : (44)
If is strictly positive, but su¢ ciently small, then government spending raises in ‡ation, which generates an even larger multiplier.

In the limit as tends to zero, nominal wages grow at rate R and the real interest rate is therefore equal to R . In that case, government spending cannot a¤ect the capitallabor ratio, by (16). Instead, a rise in g increases aggregate demand, which raises the demand for labor. But, to keep the capital-labor ratio constant, the capital stock must rise. So investment also rises. Hence, even when in ‡ation does not respond to government spending, the multiplier is larger than 1. With a decreasing marginal utility of wealth, i.e.

" > 0, this e¤ect is ampli…ed as the increase in the capital stock reduces the marginal utility of wealth, which raises the demand for consumption. If wages are not completely rigid, then the increase in labor demand raises in ‡ation. This spurs the demand for both consumption and investment, which further increases output.

However, a sizeable …scal multiplier does not necessarily imply that the implementation of a …scal stimulus is welfare enhancing.

Proposition 5 In the secular stagnation equilibrium, in the limit as tends to zero, i.e. when wages are completely rigid, a rise in government spending is welfare enhancing if and only if:

29 K + R 1 v 0 (L) v 0 (l) + c " " u > [c + g] v 0 (L) v 0 (l) ; ( 45
)
where v 0 (L) =v 0 (l) 1 for all L l.

the resulting policy can be seen as the joint implementation of the last two policies: a helicopter drop of money together with the …scal stimulus …nanced from lump-sum taxes. 29 This condition holds for any value of g, where c, l, L, and K implicitly are functions of g. It follows that the optimal value of g is such that the two sides of the inequality are equal to each other.

A rise in government spending a¤ects welfare through three channels. First, to meet the higher level of demand, employment L rises. Agents work more, which reduces their welfare. Second, when wages are rigid, the capital-labor ratio is constant and the capital stock must therefore rise. This raises the wealth of households, net of government liabilities, which is welfare enhancing. Finally, if " > 0, the increase in wealth raises the demand for consumption, which is also welfare enhancing. However, even when g = 0, it is not clear whether the positive e¤ects dominate.

If labor supply is completely inelastic, then v 0 (L) =v 0 (l) = 0 for all L < l. This implies that working more hours whenever L < l is not costly. Hence, increasing g always raises welfare as long as L < l. This yields the following corollary.

Corollary 1 If labor supply is completely inelastic, then the optimal …scal stimulus eliminates the gap between labor demand L and labor supply l.

It is important to realize that, even if " = 0, i.e. even if consumption is independent of g, eliminating under-employment is welfare enhancing as it raises the wealth of households while the disutility from supplying labor up to l is negligible.

If labor supply is in…nitely elastic, then v 0 (L) =v 0 (l) = 1. In that case, under the mild condition that c [1 " =" u ] > K, the implementation of any …scal stimulus is detrimental to welfare, despite a multiplier that is larger than 1. By continuity, this result must also hold for a su¢ ciently high elasticity of labor supply.

Corollary 2 If in the secular stagnation equilibrium c [1 " =" u ] > K and if labor supply is highly elastic, then the implementation of a …scal stimulus reduces welfare.

Thus, starting from a positive level of government spending, it would actually be desirable to cut government expenditures. This would allow workers to enjoy even more leisure.

Arguably, for plausible calibrations, v 0 (L) =v 0 (l) is unlikely to be very close to zero, even when g = 0. For instance, a severe depression, where labor demand is only 75% of labor supply, together with a Frisch elasticity of labor supply of 0.5 implies that v 0 (L) =v 0 (l) = (0:75) 1=0:5 = 0:5625. Of course, the case for a …scal stimulus is stronger if wages are not completely sticky. However, the absence of outright de ‡ation in almost any country during the Great Recession suggests that, in practice, in ‡ation is not very responsive to government spending. All this suggests that, under secular stagnation, the case for a …scal stimulus is, at best, weak.

My analysis of the …scal stimulus so far assumes that the economy remains in the secular stagnation equilibrium. However, an increase in government spending raises the real interest rate of the frictionless equilibrium. 30 Thus, a …scal stimulus that is su¢ ciently large to raise the frictionless interest rate from to makes the frictionless equilibrium consistent with the in ‡ation ceiling . Note that, to implement this equilibrium, the money supply must be growing at rate .

Of course, this policy does not eliminate the secular stagnation equilibrium. The di¢ culty is to make sure that the economy converges to the frictionless equilibrium. If this does not spontaneously occur, the government can eliminate the secular stagnation equilibrium by committing to prevent in ‡ation from falling below R . Indeed, at any point in time, it can always spend su¢ ciently to eliminate the gap between labor demand L and labor supply l, which raises in ‡ation to at least R , by (36). In theory, if this commitment is fully credible, households expect the economy to converge to the frictionless steady state with in ‡ation . In practice, it might be very di¢ cult for a government to credibly commit to spend su¢ ciently to prevent the occurrence of a steady state with underemployed workers. 31

Optimal Policy

To lift the economy out of secular stagnation, we clearly need to look beyond the usual set of policy instruments used for macroeconomic stabilization. Hence, I now characterize the optimal policy under a much richer set of taxes and subsidies. I proceed in two steps. First, I determine the optimal allocation of resources. I then solve for the …scal policy that implements this allocation in the decentralized economy.

Optimal Allocation

Wealth can either be held in the form of physical capital K t , bonds b t , or money m t . Hence, in equilibrium, the private wealth of households net of government liabilities must be equal to K t . The planner's problem is to maximize the welfare of the representative household:

Z 1 0 e t [u (c t ) v (l t ) + h (m t ) + (K t )] dt; (46)
subject to the resource constraint given by the capital accumulation equation:

_ K t = F (K t ; l t ) K t c t ; (47)
where I have used the fact that, in a …rst-best allocation, the labor used in production L t must be equal to the labor supplied by workers l t . By the maximum principle, for any 31 In the late 1930s, Hansen (1939) believed that the U.S. economy was trapped into a secular stagnation equilibrium. A possible interpretation of the subsequent history is that the military spending of World War II allowed the economy to escape secular stagnation, while the build-up of the welfare state after WWII may have been su¢ cient to allow the economy to produce at full capacity. initial stock K 0 of capital, the …rst-best allocation (c t ; K t ; l t ; m t ) is characterized by:32

_ K t = F (K t ; l t ) K t c t ; (48) _ c t c t = F K (K t ; l t ) + 0 (K t ) u 0 (c ) u 0 (c t ) u 00 (c t ) c t ; (49) v 0 (l t ) = F L (K t ; l t ) u 0 (c t ) ; (50) h 0 (m t ) = 0; (51)
together with the transversality condition:

lim t!1 e t u 0 (c t ) K t = 0: (52)
Money has no impact on the allocation of resources, but is valued by consumers who derive utility from holding money. Hence, the …rst-best allocation satiates individuals with real money balances. These equations correspond to the frictionless equilibrium of the decentralized economy where consumption is given by ( 8) with r t = F K (K t ; l t ) and a t (b t + m t ) = K t , labor supply by ( 9) with w t = F L (K t ; l t ), and where aggregate supply F (K t ; l t ) is equal to aggregate demand c t + K t + _ K t . Also, given the money demand equation ( 10), to have h 0 (m t ) = 0 in a decentralized economy, we need to have i t = 0, i.e. the Friedman (1969) rule needs to hold. This establishes the following result.

Proposition 6

The frictionless equilibrium with i t = 0 does implement the …rst-best allocation of resources.

We have previously established that, under mild conditions, the frictionless steady state is unique and stable. We can therefore consider that the …rst-best allocation converges to that steady state.

Let r n t denote the natural real interest rate of the frictionless equilibrium, de…ned as:

r n t = F K (K t ; l t ) : (53)
Recall that, to have secular stagnation with 0, the steady state natural real interest rate must be negative, i.e. r n = < 0. This implies that, in the …rst-best steady state, the capital stock must be above the golden rule level. However, this is fully e¢ cient as households derive utility from holding real wealth.

Implementation in the Decentralized Economy

For simplicity, …rms are assumed to rent capital owned by households. As investment is always positive in the …rst-best allocation that I want to implement, I assume that the price of capital is always equal to that of consumption goods, i.e. p K t = P K t =P t = 1. A capital income tax K t is imposed on the income from physical capital R t K t net of a depreciation allowance ' t P t K t , where, as before, R t denotes the nominal rental cost of capital. 33 Under full depreciation allowance, we have ' t = 1. There is an investment subsidy s I t . As we shall see, this reduces the e¤ective price of capital at time t to P t 1 s I t . The e¤ective nominal wealth of a household is therefore equal to:

A t = 1 s I t P t K t + B t + M t : ( 54
)
This wealth is taxed at rate W t at each point in time. 34 Finally, households are subject to a labor income tax L t and to a consumption tax C t . This …scal policy implies that the nominal wealth P t K t +B t +M t of the representative household evolves according to:

P t _ K t + _ P t K t + _ B t + _ M t (55) = i t B t + R t K t K t [R t K t ' t P t K t ] + s I t h P t _ K t + P t K t i P t K t + _ P t K t W t
1 s I t _ K t _ s I t K t + _ b t + _ m t = i t t W t 1 s I t K t + b t + m t (56) + 1 K t R t =P t + K t ' t 1 s I t _ s I t 1 s I t (i t t ) K t i t m t + 1 L t w t l t + t 1 + C t c t :
This expression does con…rm that the e¤ective real price of capital is 1 s I t . By arbitrage, the returns from holding capital should be identical to the returns from holding bonds. Thus, the second term on the right hand side of (56) must be equal to zero, which yields the following relationship between the real interest rate and the user cost of capital:

r t = 1 K t R t =P t + K t ' t _ s I t 1 s I t : (57)
The wealth accumulation equation simpli…es to:

_ a t = r t W t a t i t m t + 1 L t w t l t + t 1 + C t c t ; (58)
where a t = A t =P t . The corresponding no-Ponzi condition is:

lim t!1 e R t 0 (rs W s )ds a t 0: (59)
Finally, households'intertemporal utility is given by:

Z 1 0 e t " u (c t ) v (l t ) + h m t 1 + C t + a t b s t + m s t s I t K t 1 + C t !# dt; ( 60
)
where I assume that they care about the value of money and wealth at consumer prices, including consumption taxes. The sum of the wealth of the private and of the public sector must always be equal to the stock of physical capital. Hence, the government liabilities at t are equal to b s t + m s t s I t K t , where b s t , m s t , and K t are aggregate quantities that are outside the control of any single household.

The consumer's problem is to maximize utility, (60), subject to the budget constraint, (58) and ( 59), with a 0 given. The solution to the problem is characterized by:

_ c t c t = " r t W t _ C t 1 + C t + 1 u 0 (c t ) 0 a t b s t + m s t s I t K t 1 + C t !# u 0 (c t ) u 00 (c t ) c t ; (61) v 0 (l t ) = 1 L t 1 + C t w t u 0 (c t ) ; ( 62
)
h 0 m t 1 + C t = i t u 0 (c t ) ; (63)
together with the transversality condition:

lim t!1 e t u 0 (c t ) 1 + C t a t = 0: (64)
The reason why _ C t appears in the Euler equation ( 61) is that a rising path of the consumption tax makes future consumption expansive relative to current consumption. This reduces the e¤ective real interest rate faced by consumers, which raises the demand for consumption.

On the supply side, the constant returns to scale production function implies that, in equilibrium, the real rental cost of capital and the real wage are given by:

R t P t = F K (K t ; L t ) ; (65)
w t = F L (K t ; L t ) ; (66)
respectively.

The downward wage rigidity is the same as before, summarized by (36). As the …rst-best allocation is characterized by full employment, i.e. L t = l t , it cannot be implemented with in ‡ation below R . Thus, as we shall see, the …rst-best allocation can be implemented for any given path of in ‡ation such that t 2 R ;

. This requires the money supply to grow at rate t at time t.

In the …rst-best allocation, households have a zero marginal utility of real money balances, by (51). The decentralized money demand equation (63) therefore implies that, to implement the optimal allocation, we must have i t = 0 and, hence, r t = t . Fundamentally, in the laissez-faire economy, the lower bound on the real interest rate, r t , creates two distortions that need to be corrected. First, by (8), it reduces the demand for consumption. Second, by (16), it reduces the capital-labor ratio.

To correct the …rst distortion, the net-of-tax interest rate r t W t

_ C t = 1 + C t = t W t _ C t = 1 + C t
in the demand for consumption (61) must be equal to the natural real interest rate r n t . As in practice the consumption tax cannot be increasing forever, I impose that it must be constant over time, i.e. _ C t = 0. To implement the …rst-best allocation for a given t 2 R ;

, the wealth tax must therefore be given by:

W t = r n t t : (67)
This guarantees that the consumption Euler equation of the decentralized economy (61) coincides with the corresponding …rst-order condition to the planner's problem (49). Secular stagnation results from the fact that in steady state r n t = r n < , which implies that W t = r n t > t 0. To correct the second distortion, the marginal product of capital of the frictionless equilibrium F K (K t ; l t ) = + r n t must be made consistent with real interest rate of the decentralized economy r t = t . Hence, by the arbitrage relationship (57), the capital income tax K t , the depreciation allowance ' t , and the investment subsidy s I t must be jointly chosen such that:

t = 1 K t [ + r n t ] + K t ' t _ s I t 1 s I t : (68)
Let us consider three special cases of interest. For simplicity, I focus on the optimal policy with constant in ‡ation 2 R ; and in steady state, where r n t = r n = . First, relying exclusively on the investment subsidy, i.e. K = 0, we have:

s I = r n ; (69)
which is strictly positive, as r n < and > = +r n = F K (K ; l ) > 0. Second, relying on the capital income tax with full depreciation allowance, i.e. s I = 0 and ' = 1, we obtain:

K = r n r n ; (70)
which is strictly positive (to the extent that 0 implies r n < 0). In fact, the negative natural real interest rate implies that the returns from capital in the frictionless equilibrium are negative. Hence, with full depreciation allowance, a positive capital income tax absorbs some of the losses from capital. It therefore subsidizes capital. Third, relying again on the capital income tax but with no depreciation allowance, i.e. s I = 0 and ' = 0, we have:

K = r n + + r n ; (71)
which, in steady state, is negative as r n + = + < 0. With no depreciation allowance, the capital income tax needs to be negative to subsidize capital. Importantly, these three policies are equivalent as they all implement the same marginal product of capital r n + = F K (K ; l ).

The optimality condition for labor supply (50) implies that, if both consumption and the capital-labor ratio are at their …rst-best levels, then so is the supply of labor. Hence, by (62) and (66), the labor income and consumption taxes can both be set equal to zero, i.e. L t = C t = 0. Let us now determine the level of the lump-sum transfers t that balances the intertemporal government budget constraint. The accumulation of government liabilities is given by:

_ b s t + _ m s t = r t [b s t + m s t ] i t m s t + t + s I t h _ K t + K t i K t [R t =P t ' t ] K t W t a t : (72)
Rearranging terms, this can be written as:

_ b s t + _ m s t s I t _ K t _ s I t K t = r t W t b s t + m s t s I t K t (73) i t m s t + t + s I t [r t + ] K t _ s I t K t K t [R t =P t ' t ] K t W t K t :
But, by the arbitrage relationship (57), we know that

s I t [r t + ] _ s I t K t [R t =P t ' t ] = r t + R t =P t .
As the sum of the wealth of the private and of the public sector must be equal to the stock of physical capital, the government is always solvent, and able to redeem the money supply, provided that it satis…es the following no-Ponzi condition:35

lim t!1 e R t 0 (rs W s )ds b s t + m s t s I t K t 0: (74)
Hence, the government's intertemporal budget constraint is given by:

b 0 + m 0 s I 0 K 0 (75) + Z 1 0 e R t 0 (rx W x )dx i t m s t + t + (r t + R t =P t ) K t W t K t dt 0;
But, in the …rst-best allocation, i t = 0. Also, the optimal policy for a given t 2 R ; is such that r t = t , R t =P t = r n t and W t = r n t t . It immediately follows that the government budget constraint simpli…es to:

b 0 + m 0 s I 0 K 0 + Z 1 0 e R t 0 r n x dx t dt 0: (76)
This shows that, if s I 0 = 0, the optimal policy is self-…nancing as the wealth tax pays for the subsidization of income from physical capital. The only e¤ect of the policy is to modify the real interest rate at which future lump-sum transfers are discounted.

The initial investment subsidy s I 0 reduces the initial price of capital, which tightens the budget constraint of households. This mechanically relaxes the government budget constraint. Hence, s I 0 acts as a levy on initial capital. However, if the natural real interest rate r n t is eventually negative, then in…nitesimally small negative lump-sum transfers t forever are su¢ cient to balance the government budget constraint for any …nite value of s I 0 . In that case, the lump-sum transfers that the government can a¤ord are not a¤ected by s I 0 . Proceeding as before, cf. (41), it is straightforward to check that, under the optimal policy, the transversality conditions of the household (64) and of the planner (52) coincide.

We have therefore characterized by construction a policy that can implement the …rstbest allocation for any path of in ‡ation such that t 2 R ; at all time t. The main result is summarized in the following proposition. Proposition 7 At any time T , the …rst-best allocation can be implemented in the decentralized economy for any given path of in ‡ation (^ t ) [START_REF] Acemoglu | Introduction to Modern Economic Growth[END_REF] t=T such that ^ t 2 R ; for all t T by setting the money growth rate t , the tax on wealth W t , the capital income tax K t , the depreciation allowance ' t , and the investment subsidy s I t such that:

t = ^ t ; (77)
W t = r n t ^ t ; (78)
1 s I t [ ^ t ] = 1 K t [ + r n t ] + K t ' t _ s I t : (79)
This policy is revenue neutral, except that s I 0 acts as a levy on initial capital.

The optimal policy implements the allocation of the frictionless equilibrium with i t = 0. However, it implements this allocation with in ‡ation t = ^ t 2 R ; , rather than the much higher rate t = r n t . By construction, the optimal policy is consistent with the in ‡ation ceiling at all time, except possibly when the policy is implemented. Indeed, as the economy jumps from the secular stagnation to the frictionless equilibrium, the zero lower bound ceases to bind. 36 If the policy is implemented at time T with i t = 0, then the price level at T must be given by:

P T = M T m .
To prevent the price level from jumping upon implementation of the policy, the money supply at time T must be equal to mP T " , where P T " denotes the price level that prevailed just before. If the real money supply at time T " was larger than the transaction demand for money, i.e. m T " > m, the government must redeem a quantity m T " m of money. This can generate a massive increase in government debt. 37 However, in theory, with a permanently negative natural real interest rate, the government's no-Ponzi condition (74) is satis…ed for an arbitrarily large initial level of debt, provided that the primary budget is subsequently balanced. These considerations nevertheless show that, if the government used to rely excessively on the printing press to …nance its expenditures, then the sustainability of public debt can become an important issue when the optimal policy is implemented. For simplicity, let us consider that the government targets a constant rate of in ‡ation

^ 2 R ;
and that, once the economy is in steady state, the policy becomes time invariant, i.e. t = , W t = W , K t = K , ' t = ', and s I t = s I . Importantly, the optimal policy does not eliminate the secular stagnation equilibrium, where in ‡ation is below the target ^ . 38 There is however one important exception.

Proposition 8

The frictionless and secular stagnation steady state equilibria can coincide if and only if the optimal policy is implemented with in ‡ation target ^ = R . If the secular stagnation equilibrium is unique, which must be the case for su¢ ciently close to zero, then the optimal policy with ^ = R uniquely implements the …rst-best allocation.

Implementing the optimal allocation with in ‡ation target ^ = R requires the most aggressive policy intervention, i.e. the largest wealth tax and the highest subsidization of physical capital. Indeed, a massive intervention is necessary to make the optimal allocation consistent with an in ‡ation rate as low as R .

If the government would like to be in the optimal allocation with in ‡ation , then it should …rst implement the optimal policy with ^ = R in order to escape secular stagnation. It should then raise the growth rate of the money supply to , while reducing the size of the wealth tax, and of the subsidization of capital that it …nances, until the nominal interest rate is back to zero.

The wealth of households is composed of physical capital, bonds, and money. A potential concern with the optimal policy is that money might be harder to tax than physical capital or bonds. Households might indeed be tempted to accumulate stockpiles of cash to avoid paying the wealth tax. But, untaxed money yields a real interest rate t , which, under the in ‡ation ceiling, is higher than the steady state natural real interest rate. By arbitrage, this would induce the after-tax real interest rate on physical capital and on bonds to be above the natural rate, which would prevent the implementation of the optimal allocation. It is therefore essential that the government is able to tax money.

To …ght against money laundering, many countries already enforce signi…cant restrictions that reduce the value of large sums of undeclared currency. There are limits to the size of payments than can be made in cash. 39 Banks cannot accept large cash deposits if the origin of the funds is not proven. Controls can also be made when old bills occasionally become obsolete and are replaced by new ones. If these measures are not su¢ cient, then the government can adopt a payment system that exclusively relies on electronic money.

Note that, if money is straightforward to tax, then an alternative way to implement the …rst-best allocation is to alleviate the zero lower bound on the nominal interest rate. For this, a government targeting in ‡ation ^ t only needs to tax money at rate W t , while setting the money growth rate t and the tax rate on money W t according to (77) and (78), respectively. The lower bound on the nominal interest rate becomes W t , which is negative. In that case, when the nominal rate is at its lower bound, the real interest rate is at its natural level, as i t = W t = r n t + ^ t implies r t = i t ^ t = r n t . This policy is revenue neutral.

In the absence of electronic money, the tax on wealth seems to o¤er a more reliable way to reduce the net-of-tax real interest rate faced by households than the tax on money. Indeed, the wealth tax is e¤ective provided that the authorities can prevent large accumulations of undeclared currency. By contrast, if real money balances are small and the tax on money is poorly enforced on small amounts, then it is not clear that, by arbitrage, it can reduce the net-of-tax real interest rate faced by households on their holdings of physical capital and bonds. This suggests that, without electronic money, it is likely to be very di¢ cult to enforce negative nominal interest rates.

Heterogeneity

For simplicity, I have so far relied on a representative household model. However, my results only require a strictly positive mass of households to have a preference for wealth. Indeed, in the presence of heterogeneity, impatient households borrow from those who are more patient, until they reach their borrowing limit. Once this limit is reached, the real interest rate must be su¢ ciently low to induce the most patient households to consume enough to allow the economy to produce at full capacity. In other words, in steady state, the natural real interest rate must be determined by the behavior of the most patient households, i.e. by those who have a preference for wealth.

Under a dynastic interpretation of the in…nitely lived households, the borrowing limit should be determined by the impossibility of bequeathing negative wealth. For instance, in a discrete time model where one period corresponds to one generation, households face a zero debt limit. As the borrowing limit must eventually be binding, I consider a very simple form of heterogeneity whereby a fraction of households have a preference for wealth, while the remaining fraction consists of hand-to-mouth consumers who do not derive any utility from holding wealth.

Interestingly, in this setup, if the real interest rate is excessively high, then the economy can only produce at full capacity if poor households accumulate an excessive amount of debt. But, once the rich lenders realize that debt cannot be passed on to the next generation of poor households, the economy has to go through a painful deleveraging episode. As pointed out by Summers (2014), the fact that the credit boom that occurred before the Great Recession did not overheat the U.S. economy is fully consistent with the secular stagnation hypothesis.

In the presence of heterogeneity, the government might want to redistribute resources across households. In general, as in Correia, Farhi, Nicolini and Teles (2013), the set of implementable allocation is not modi…ed by the zero lower bound. Indeed, the government can always implement its desired redistribution policy, while using wealth taxes and investment subsidies as described in the previous section to overcome the zero lower bound. 40 In the context of secular stagnation, there is however one important exception where the zero lower bound can allow the government to implement an allocation that would not otherwise be feasible. This occurs when the government would like to redistribute to hand-to-mouth households, but cannot rely on household-speci…c lump-sum taxes or subsidies and does not want to modify the tax rate on wealth, as this would distort the consumption behavior of wealthy households. 41 Hence, the government can only raise the consumption of the hand-to-mouth by giving a lump-sum subsidy t to all households. If t is large, this clearly violates the conservative government budget constraint, given by:

b 0 + m 0 + Z 1 0 e R t 0 rxdx [ t i t m t ] dt 0: (80)
But, if the government does not need to redeem the outstanding money supply, the budget constraint becomes:

b 0 + Z 1 0 e R t 0 rxdx [ t t m t ] dt 0: (81)
The subsidy can be …nanced by printing money forever. The wealth of rich households must therefore grow without bounds. Indeed, if the conservative version of the government's no-Ponzi condition is violated and if markets clear, then, by Walras' law, the no-Ponzi condition of rich households must be slack, for any value of the real interest rate. The increase in wealth is not distortionary, i.e. it does not a¤ect the behavior of rich households, provided that it does not change their marginal utility of wealth. This either occurs if they consider money as a government liability, or if they have a constant marginal utility of wealth. Thus, in the absence of household speci…c lump-sum taxes, the only way to redistribute to the poor without increasing the tax rate on wealth is to print money. This policy can be sustained forever provided that the wealthy have a zero marginal propensity to consume out of extra money supplied. 42 Paradoxically, this egalitarian allocation of consumption can only be implemented by making the rich in…nitely wealthy. This theoretical possibility is speci…c to secular stagnation where the zero lower bound is potentially binding forever. Needless to emphasize that such policy is very dangerous should the economy ever escape secular stagnation. Hence, crucially, the increase in the consumption of the hand-to-mouth should not induce the economy to produce at full capacity, as in ‡ation would then explode.

Conclusion

In this paper, I have relied on a Ramsey model with money to o¤er a simple theory of secular stagnation. The permanent failure of the economy to produce at full capacity results from three plausible assumptions: (i) the central bank has a low in ‡ation ceiling; (ii) a positive mass of households has a very high propensity to save, due to a preference for wealth; (iii) wages are downward rigid. Despite the simplicity of this theory, I was able to derive the main Keynesian paradoxes.

I have argued that traditional monetary and …scal policies are not e¢ cient in the context of a permanent liquidity trap. To escape secular stagnation without raising the in ‡ation ceiling, the government should instead simultaneously tax wealth and subsidize income from physical capital, either through an investment subsidy or a reduction in the taxation of corporate income. This policy is revenue neutral and implements the …rst-best allocation of resources.

When the economy escapes the liquidity trap, households cease to rely on money for saving. To prevent the price level from jumping upon implementation of the optimal policy, the government must therefore shrink the money supply to the level of the transaction demand for money. This can signi…cantly raise the debt-to-GDP ratio. In theory, an arbitrarily large amount of public debt is sustainable when the real interest rate is negative forever. But, in practice, a country that used to heavily rely on monetary …nancing of public de…cits might be reluctant to implement the optimal policy. Hence, an excessive reliance on the printing press can eventually result in a country choosing to remain permanently trapped into secular stagnation.

B Proof of Lemma 2

As in the proof of the previous lemma, let k = K=L and f (k) = F (k; 1). To prove the lemma, we compare the slope of the aggregate demand and of the aggregate supply curves. For this, we …rst need to compute the derivative of c, k, and l with respect to r, where r = i with i 0. Recall that, when 2 R ; , the labor market clears, i.e.

L = l.

From the equation for the demand for capital (16), we have:

dk dr = 1 f 00 (k) < 0:
Implicitly di¤erentiating the optimality condition for labor supply ( 9) with w = f (k) kf 0 (k) yields:

dl dr = 1 v 00 (l)
kf 00 (k) u 0 (c) dk dr + wu 00 (c) dc dr ;

= 1 v 00 (l)
ku 0 (c) + wu 00 (c) dc dr :

Finally, from the optimality condition for consumption (8) with _ c t = 0 and a t (b t + m t ) = K = kl, we have:

dc dr = 1 u 00 (c) 0 (kl) ( r) 2 + 00 (kl) r l dk dr + k dl dr ; = u 0 (c) u 00 (c) u 0 (c) 0 (kl) + 00 (kl) 0 (kl) l f 00 (k) + k v 00 (l)
ku 0 (c) + wu 00 (c) dc dr ;

= u 0 (c) u 00 (c) u 0 (c) 0 (kl) kl 00 (kl) 0 (kl) ku 0 (c) lv 00 (l) 1 kf 00 (k) + kl 00 (kl) 0 (kl) v 0 (l) lv 00 (l) dc dr :
where I have used the fact that r = 0 (kl) =u 0 (c) to derive the second line and that wu 0 (c) = v 0 (l) to derive the third line. Rearranging terms, this expression can be written as:

dc dr = 1 lv 00 (l) v 0 (l) + kl 00 (kl) 0 (kl) lv 00 (l) v 0 (l) u 0 (c) u 00 (c) u 0 (c) 0 (kl) + kl 00 (kl) 0 (kl) ku 0 (c) lv 00 (l) + 1 kf 00 (k) ;
which shows that consumption is unambiguously decreasing in the real interest rate.

Substituting this expression into the derivative of labor supply yields:

dl dr = ku 0 (c) v 00 (l) + l lv 00 (l) v 0 (l) + kl 00 (kl) 0 (kl) u 0 (c) 0 (kl) + kl 00 (kl) 0 (kl) ku 0 (c) lv 00 (l) + 1 kf 00 (k) ; = l lv 00 (l) v 0 (l) + kl 00 (kl) 0 (kl) u 0 (c) 0 (kl) lv 00 (l) v 0 (l) ku 0 (c) lv 00 (l) + kl 00 (kl) 0 (kl) 1 kf 00 (k) ; = 1 lv 00 (l) v 0 (l) + kl 00 (kl) 0 (kl) lu 0 (c) 0 (kl) 1 k 0 (kl) v 0 (l) + kl 00 (kl) 0 (kl) l kf 00 (k) :
But, as 0 (kl) = ( r) u 0 (c) = ( f 0 (k) + ) u 0 (c) and v 0 (l) = wu 0 (c), we have:

dl dr = 1 lv 00 (l) v 0 (l) + kl 00 (kl) 0 (kl) lu 0 (c) 0 (kl) 1 k ( f 0 (k) + ) w + kl 00 (kl) 0 (kl) l kf 00 (k) ; = 1 lv 00 (l) v 0 (l) + kl 00 (kl) 0 (kl) lu 0 (c) 0 (kl) f (k) k ( + ) w + kl 00 (kl) 0 (kl) l kf 00 (k)
;

where I have used the fact that w = f (k) kf 0 (k). Clearly, if labor supply is su¢ ciently inelastic, then dl=dr can be arbitrarily close to zero. Let y s and y d denote the aggregate supply and the aggregate demand, respectively. We have y s = F (K; L) = F (k; 1) L = f (k) l, since L = l, and y d = c + K = c + kl. We can now compare the slopes of the aggregate demand and supply curves:

d y s y d dr = d ([f (k) k] l c) dr ; = [f (k) k] dl dr + [f 0 (k) ] l dk dr dc dr ; = [f (k) k] dl dr + rl f 00 (k) dc dr : (B1)
where, to get the third line, I have used the fact that f 0 (k) = r. If labor supply is inelastic, i.e. dl=dr = 0, we have:

d y s y d dr = rl f 00 (k) dc dr l f 00 (k) dc dr ;
where I have used the fact that r = 0 (kl) =u 0 (c) 0. Hence, if = 0:

d y s y d dr dc dr > 0;
or, equivalently: dy s dr > dy d dr :

But, recall that we have r = i > 0 = r n and that y s = y d when r = r n . The inequality therefore implies that y d is strictly smaller than y s for all r > r n . By continuity, this must be true provided that is su¢ ciently close to zero and that labor supply is su¢ ciently inelastic.

Even if labor supply is very elastic, the …rst term of (B1) is positive under mild conditions, which strengthens the above result. More formally, we have:

[f (k) k] dl dr = y s y d + c l 1 lv 00 (l) v 0 (l)
kl 00 (kl) 0 (kl)

y s y d + c kl ( + ) w + l kf 00 (k) kl 00 (kl) 0 (

C Proof of Lemma 3

As in the proof of the previous two lemmas, let k = K=L and f (k) = F (k; 1). By ( 8), ( 9), . By assumption, y d < y s for = R . Thus, to prove the existence of an equilibrium, it is su¢ cient to show that, in the limit as tends to R , we have y d > y s . Let f (k) = f (k) kf 0 (k). We know that, for any K and L, F L (K; L) = f (K=L) K=Lf 0 (K=L) = f (K=L). We also have F (0; L) = 0 for any L, which implies that F L (0; L) = 0. Thus, F L (0; L) = f (0=L) = f (0) = 0. We also have f 0 (k) = kf 00 (k), which shows that f (k) is strictly increasing for any k > 0. Hence, the unique solution to f (k) = 0 for k 0 is k = 0.

= f 0 (k) , w = f (k) kf 0 (k), and (36)
To ensure that the demand for consumption, given by ( 8), is strictly positive when = R

, we need to have + = + R > 0 or, equivalently, < R + . This in fact guarantees that consumption is positive for all 2 R ; R . Thus, by ( 8), = f 0 (k) , w = f (k) kf 0 (k), and ( 9), the values of c, k, w, and l are strictly positive and …nite when = R . The remaining variable, L, is implicitly determined 46 Recall that, by footnote 12, < . This guarantees that = f 0 (k) de…nes k as a continuous function of for all 2 R

; . Also, note that the consumer's problem implies that we have a corner solution with c = 0 and v 0 (l) = +1 whenever . However, this case does not arise when < R + since > R > .

Totally di¤erentiating these last two equations with respect to r yields:

dk dr = 1 f 00 (k) ; dL dr = 1 f (k) k dc dr L f 0 (k) f (k) k dk dr ;
respectively. It follows that:

k dL dr + L dk dr = k f (k) k dc dr Lk f 0 (k) f (k) k dk dr + L dk dr ; = k f (k) k dc dr + L f (k) kf 0 (k) f (k) k dk dr ; = kL c dc dr + L 2 c f (k) f 00 (k) ;
where I have used the fact that f (k) k = c=L. Totally di¤erentiating u 0 (c) = 0 (kL) = ( r) yields:

dc dr = 1 u 00 (c) 0 (kL) ( r) 2 + 00 (kL) r k dL dr + L dk dr ; = 0 (kL) u 00 (c) ( r) " u 0 (c) 0 (kL) + 00 (kL) 0 (kL) " kL c dc dr + L 2 c f (k) f 00 (k) ## ; = u 0 (c) cu 00 (c) " cu 0 (c) 0 (kL) + kL 00 (kL) 0 (kL) L f (k) kf 00 (k) # + u 0 (c) cu 00 (c)
kL 00 (kL) 0 (kL) dc dr ;

where I have used the fact that 0 (kL) = ( r) = u 0 (c). It follows that:

dc dr = 1 " " u " cu 0 (c) 0 (kL) " L f (k) kf 00 (k) # ;
where " = kL 00 (kL) = 0 (kL) and " u = cu 00 (c) =u 0 (c). Thus, dc=dr < 0 if and only if " < " u . We therefore have:

k dL d + L dk d = kL c " dc dr + L k f (k) f 00 (k) # ; = kL c 1 " " u " cu 0 (c) 0 (kL) " L f (k) kf 00 (k) + (" " u ) L k f (k) f 00 (k) # ; = kL c 1 " " u " cu 0 (c) 0 (kL) " u L k f (k) f 00 (k) # :
with respect to , , %, , and g. I set = 1, except in the proof of Proposition 2, % = = 1, except in the proof of Proposition 3, and g = 0, except in the proof of the Proposition 4 and 5.

Finally, throughout the proofs, I will use the notations " u = cu 00 (c) =u 0 (c) and " = K 00 (K) = 0 (K).

E.1 Proof of Proposition 1

Totally di¤erentiating (E2) and (E5) with respect to yields:

dk d = 1 f 00 (k) d d ; dL d = 1 f (k) k dc d L f 0 (k) f (k) k dk d ;
respectively. It follows that:

k dL d + L dk d = k f (k) k dc d Lk f 0 (k) f (k) k dk d + L dk d ; = k f (k) k dc d + L f (k) kf 0 (k) f (k) k dk d ; = kL c dc d L 2 c f (k) f 00 (k) d d ;
where I have used the fact that f (k) k = c=L. Totally di¤erentiating (E1) yields:

dc d = 1 u 00 (c) 0 (kL) ( + ) 2 d d + 00 (kL) + k dL d + L dk d ; = 0 (kL) u 00 (c) ( + ) " u 0 (c) 0 (kL) d d + 00 (kL) 0 (kL) " kL c dc d L 2 c f (k) f 00 (k) d d ## ; = u 0 (c) cu 00 (c) " cu 0 (c) 0 (kL) + kL 00 (kL) 0 (kL) L f (k) kf 00 (k) # d d + u 0 (c) cu 00 (c) kL 00 (kL) 0 (kL) dc d ;
where I have used the fact that 0 (kL) = ( + ) = u 0 (c). It follows that:

dc d = 1 " u " " cu 0 (c) 0 (kL) " L f (k) kf 00 (k) # d d ;
where " u = cu 00 (c) =u 0 (c) and " = kL 00 (kL) = 0 (kL). Hence:

k dL d + L dk d = kL c " dc d L k f (k) f 00 (k) d d # ; = kL c 1 " u " " cu 0 (c) 0 (kL) " L f (k) kf 00 (k) (" u " ) L k f (k) f 00 (k) # d d ; = kL c 1 " u " " cu 0 (c) 0 (kL) " u L k f (k) f 00 (k) # d d :
Di¤erentiating (E6) with respect to gives:

dy d = dc d + k dL d + L dk d ; = 1 " u " " cu 0 (c) 0 (kL) " L f (k) kf 00 (k) + kL c " cu 0 (c) 0 (kL) " u L k f (k) f 00 (k) ## d d ; = 1 " u " " c + kL c cu 0 (c) 0 (kL) " + kL c " u L f (k) kf 00 (k) # d d :
If " u " > 0, then the term in the main square bracket is strictly positive. In that case, a fall in in ‡ation results in a lower output level. Totally di¤erentiating (E3) with respect to yields:

f (k)+ f (k) d d + R f 0 (k) dk d = f L l k + f 0 L l k L l dk d + k l dL d Lk l 2 dl d ;
where dl=d can be computed from (E4). We know that, in the secular stagnation equilibrium, 2 R ; R . Hence, as tends to zero, R also tends to zero. It follows that, as tends to zero, the above expression simpli…es to:

d d = f L l k f (k) f (k) :
But, L < l and f ( ) is a strictly increasing function. 48 Hence, d =d < 0. By continuity, this result remains true if is su¢ ciently close to zero. 48 In the limit as tends to zero, we know that L < l. Indeed, when = R , we know by Lemma 2 that y

s = lf (k) > c + kl = y d . If 2 R
; R , then, in the limit as tends to zero, c, l, or k are not a¤ected by the fact that < R (and we still have lf (k) > c + kl), while in the secular stagnation equilibrium L is such that Lf (k) = c + kL. Combining these two equations immediately yields l > L.

E.2 Proof of Proposition 2

Di¤erentiating (E3) with respect to gives:

f (k) d d + R f 0 (k) dk d = l f 0 L l k L dk d + k dL d Lk l dl d :
But, given that in the secular stagnation equilibrium 2 R ; R , we know that R tends to zero as tends to zero. Also, dc=d , dk=d , dl=d , and dL=d , which can be derived from (E1), (E2), (E4), and (E5), respectively, are only a¤ected by through d =d . Hence, as tends to zero, d =d also tends to zero.

Relying on (E2) and (E5), where does not appear, and proceeding as in the proof of Proposition 1, we obtain: where, as before, " u = cu 00 (c) =u 0 (c) and " = kL 00 (kL) = 0 (kL). It follows that: Recall that d =d tends to zero as tends to zero. 49 It unambiguously follows that dc=d < 0 and d ( kL) =d < 0, provided that " u > " .

k dL d + L dk d = kL c dc d L 2 c f (k) f 00 (k)
dc d = 1 " u " " c + " L f (k) kf 00 (k)

E.3 Proof of Proposition 3

I …rst focus on the impact of the disutility of labor % while setting = 1; I then focus on the impact of total factor productivity while setting % = 1.

E.3.1 Disutility of Labor

Relying on (E1), (E2), and (E5), where % does not appear, and proceeding as in the proof of Proposition 1, we obtain: If " u > " and if a rise in the disutility of labor is in ‡ationary, then, by the second expression, it must increase consumption c while, by the third expression, it must raise investment kL. Totally di¤erentiating (E4) with respect to % yields: As tends to zero, R also tends to zero. Thus, solving for d =d% shows that it tends to:

dk d = 1 f 00 (k)
1 f (k) f 0 L l k kL l 2
v 0 (l) %v 00 (l) , d =d < 0, which ampli…es the fall in consumption and in investment.

which is strictly positive provided that > 0.50 Thus, for su¢ ciently small but strictly positive, we have d =d% > 0. It follows that a rise in the disutility of labor increases in ‡ation, consumption, investment, and, hence, output.

E.3.2 Total Factor Productivity

Totally di¤erentiating (E2) and (E5) with respect to yields:

dk d = f 0 (k) f 00 (k) 1 f 00 (k) d d ; dL d = 1 f (k) k Lf (k) + dc d L [ f 0 (k) ] dk d ; = L 2 f (k) c + L 2 c f 0 (k) [ f 0 (k) ] f 00 (k) + L c dc d + L 2 c [ f 0 (k) ] f 00 (k) d d :
It follows that: This shows that, if " u > " and if a rise in total factor productivity is de ‡ationary, then it must reduce both consumption c and investment kL. Di¤erentiating (E4), we have: As tends to zero, R also tends to zero. Thus, solving for d =d shows that it tends to:

k dL d + L dk d = kL 2 c f (k) + kL 2 c f 0 (k) [ f 0 (k) ] f 00 (k) Lf 0 (k) f 00 (k) + kL c dc d + kL 2 c [ f 0 (k) ] f 00 (k) L f 00 (k)
1 f (k) f 0 L l k Lk l 2 f (k) u 0 (c) v 00 (l) + R kf 0 (k) ;
which is strictly negative provided that > 0. Thus, for su¢ ciently small but strictly positive, we have d =d < 0. It follows that a rise in total factor productivity reduces in ‡ation, consumption, investment, and, hence, output.

E.4 Proof of Proposition 4

Totally di¤erentiating (E2) and (E5) with respect to g yields: where I have used the fact that 0 (kL) = ( + ) = u 0 (c). Hence: This shows that d =dg tends to zero as tends to zero. Moreover, if (c + g) " u > c" , then, for strictly positive but su¢ ciently small, we have d =dg > 0.

dk dg = 1 f 00 (k)
Finally, the …scal multiplier is obtained by di¤erentiating (E6): Thus, in the limit as tends to zero, d =dg = 0 and the multiplier is equal to:

1 + kL + c " "u (c + g) c "
"u

:

When is strictly positive, but su¢ ciently small, then d =dg > 0 and the multiplier is even larger.

E.5 Proof of Proposition 5

In steady state, welfare is equal to:

W = u (c) v (L) + h ( m) + (kL) ,
where I have used the fact that, in equilibrium, the wealth of households net of government liabilities must be equal to the stock of physical capital. In the limit as tends to zero, in ‡ation is equal to R . Thus, by (E2) with = R , we know that k is independent of

Figure 1 :

 1 Figure 1: Aggregate Demand (AD) and Aggregate Supply (AS) curves

  kl) :where I have used the fact thaty s y d = [f (k) k] l c. If c > K for all r 2 [r n ; ],then this expression is positive for r = r n , since y s = y d when r = r n . This implies that dy s =dr > dy d =dr for r = r n . By induction, as r increases from r n to , we havey d < y s , [f (k)k] dl=dr > 0, and dy s =dr > dy d =dr for all r 2 [r n ; ].

  , we know that c, l, k, w, and L are continuous functions of r = for 2 R ; . 46 Hence, aggregate demand y d = c + kL and aggregate supply y s = Lf (k) are continuous functions of for 2 R ;

dl d% = 1

 1 %v 00 (l) v 0 (l) kf 00 (k) u 0 (c) dk d% + f (k) u 00 (c)

  With a Cobb-Douglas aggregate production function, we havef (k) = k & for some & 2 [0; 1]. This implies that kf (k) f 00 (k) + f 0 (k) f (k) = 0.Hence, proceeding as in the proof of Proposition 1, we obtain:

  g) f 00 (k) [[f (k) k] k [f 0 (k)

R

  l) kf 00 (k) u 0 (c) dk dg + f (k) u 00 (c) dc dg ; = c" (c + g) " u c" f (k) u 00 (c) v 00 (l) + 1 v 00 (l) " ku 0 (c) + c f (k) u 00 (c) (c + g) " u c" " (c + g) u 0 (c)Substituting the above expressions into this equation gives: also tends to zero. Thus, solving for d =dg shows that it tends to:

  kL" u + c" (c + g) " u c" + 1 (c + g) " u c" " c (c + g) u 0 (c) kL + c" =" u (c + g) c" =" u + 1 (c + g) " u c" " cu 0 (c) f (k) 0 (kL) ( kL" u + c" ) L f (k) kf 00 (k) # d dg :

Also,[START_REF] Piketty | On the Long-Run Evolution of Inheritance: France 1820-2050[END_REF] relies on a model with wealth in the utility to show that the inheritance-to-

Interestingly, in the secular stagnation equilibrium, in ‡ation is essentially determined by the downward wage rigidity. Hence, the real interest rate is typically well above its lower bound, which results in a very depressed level of aggregate demand.

They also considered the possibility that demand is depressed due to a rise in inequality, where agents with low propensity to consume become wealthier, or to a fall in the price of investment goods.

However, investment goods cannot be transformed into consumption goods, i.e. investment cannot be negative.

rxdx [ s + g s s m s s ] ds;(20)[START_REF] Caballero | The Safety Trap[END_REF] If the government can implement a discrete increase in the money supply at time 0, then it can e¤ectively choose the initial price level P 0 such as to dilute the initial stock B 0 of government debt. Thus, taking M 0 as given in the context of monetary policy is like preventing the taxation of initial capital in the context of …scal policy.

If is close to zero and R close to , then in ‡ation might have to rise above the ceiling to compensate for a fall in the marginal product of capital or labor. This can prevent the existence of an equilibrium.

[START_REF] Eggertsson | A Model of Secular Stagnation[END_REF] impose a similar wage rigidity assumption, but assume that R = 0.

Clearly, the real wage being equal to both[START_REF] Kocherlakota | Explaining the Fiscal Theory of the Price Level[END_REF] and (36), the labor market does not necessarily clear in equilibrium.

In theory, aggregate demand can also be decreasing in in ‡ation for some 2 R ; . This is due to the investment component of aggregate demand, which is equal to (K=L) L. However, this e¤ect is unlikely to dominate as the consumption component typically is strongly increasing in in ‡ation.

By totally di¤erentiating the equations that characterize the frictionless steady state equilibrium, it can be shown that a su¢ cient condition to have the capital stock K increasing in the marginal utility of wealth is that the natural real interest rate is negative (or positive but su¢ ciently close to zero) and either that labor supply is su¢ ciently inelastic or that c > K. In that case, consumption c is also increasing in the marginal utility of wealth provided that the natural real interest rate is not too negative, i.e. provided that the capital stock is not too high.

Recall that, by (9) and (15), we have v 0 (l) = F L (K; L) u 0 (c).

It can be shown that in the frictionless equilibrium, if the real interest rate is negative (or positive but su¢ ciently close to zero), c > K, and K 00 (K) = 0 (K) < cu 00 (c) =u 0 (c), then a fall in the disutility of labor raises consumption, investment, and output. Also, if the real interest rate is negative (or positive but su¢ ciently close to zero), labor supply is su¢ ciently inelastic, and K 00 (K) = 0 (K) < cu 00 (c) =u 0 (c), then a rise in total factor productivity raises output. Note that these su¢ cient conditions are far from necessary.

It can be shown that a su¢ cient condition for this is c + g K and f 0 (k) 0.

It is straightforward to check that the Hamiltonian is strictly concave in c t , l t , m t , and K t and that the corresponding current-value costate variable is equal to u 0 (c t ) and that it is therefore always strictly positive. Hence (by Acemoglu 2009, Theorem 7.14, page 257), for a given value of K 0 , the optimality conditions de…ne the unique solution to the optimal control problem.

If …rms were the owners of capital, then K t would correspond to a corporate income tax.

In the optimal allocation, the nominal return from wealth is equal to zero and the real return is negative (as, in steady state, r n < 0). Hence, in that context, it seems more natural to tax the stock of wealth rather than the (negative) income from wealth.

This is a generalization of the conservative government budget constraint, given by[START_REF] Piketty | On the Long-Run Evolution of Inheritance: France 1820-2050[END_REF].

Even if the nominal interest rate remains equal to zero, the zero lower bound is no longer preventing that rate from being negative. Hence, as explained in the derivation of the steady state equilibrium, an equilibrium where some money is used for saving is unstable.

In my model, the government corresponds to the consolidation of the ministry of …nance and of the central bank. In practice, the central bank can redeem the money supply by selling the government bonds that it previously acquired to …nance the public de…cit. This does not change the indebtedness of the ministry of …nance, but raises the amount of debt held by the private sector rather than by the central bank. In Japan, the debt-to-GDP ratio amounts to 230% of GDP, a quarter of which is held by the central bank. Thus, the amount of debt held by the private sector only amounts to 170% of GDP. If Japan escapes the liquidity trap, this ratio would need to increase sharply to avoid a jump in the price level.

Note that, the optimal policy is revenue neutral even if it fails to implement the optimal allocation.

In France, the limit is 1 000 euros for commercial transactions and 10 000 euros for real estate transactions.

Under secular stagnation, it might be tempting to stimulate aggregate demand by taxing the wealthy and giving lump-sum subsidies to hand-to-mouth households. However, if this is not optimal in the absence of the zero lower bound, then it remains suboptimal with it.

Given the limited amount of heterogeneity in the model, a non-linear wealth tax could act as a lump-sum tax. I therefore assume that this instrument is not available.

This assumption does not seem outrageous in light of the recent Japanese experience, where a 230% debt-to-GDP ratio fails to lift the economy out of secular stagnation.

The real interest rate does not directly a¤ect the other equations that characterize the secular stagnation steady state.

Using the above expressions, it is possible to show that for strictly positive, but su¢ ciently small,

When = 0, nominal wages are completely rigid and always rise at rate R . Hence, in ‡ation is always equal to R and cannot be a¤ected by anything.

A Proof of Lemma 1

Let k = K=L and f (k) = F (k; 1). This implies that F K (K; L) = f 0 (k) and F L (K; L) = f (k) kf 0 (k). Note that f (0) 0f 0 (0) = F L (0; L) = 0 as F (0; L) = 0 for any value of L.

In a frictionless equilibrium with > R , the labor market must clear, i.e. L = l. The goods market clearing condition (24) implies that:

By (9) and (15), we have:

Finally, by (8) with _ c t = 0 and a t (b t + m t ) = K = kl and by (16), we have:

These three equations jointly characterize the steady state frictionless equilibrium.

Let k be uniquely de…ned by f ( k) = k. Substituting (A1) into (A2) yields:

For any given value of k 2 (0; k), as c increases from 0 to +1, the right hand side decreases from +1 to 0 while the left hand side is increasing and non-negative. Hence, for any value of k 2 (0; k), there is a unique corresponding value of c. Moreover, di¤erentiating (A4) with respect to k reveals that c is a continuous function of k for k 2 (0; k). Also, in the limit as k tends to k, c must tend to 0. 43 Let k be uniquely de…ned by

Thus, as k tends to k from below, c tends to c. 45 For any given value of c 2 (0; c), as k increases from k to k, the right hand side of (A5)

while the left hand side is weakly decreasing and tends to lim K!+1 0 (K) = [ + f 0 ( k)]u 0 ( c) as k tends to k. Hence, for any value of c 2 (0; c), there is a unique corresponding value of k 2 ( k; k). Moreover, di¤erentiating (A5) with respect to c reveals that k is a continuous function of c for c 2 (0; c).

The steady state frictionless equilibrium is fully characterized by (A4) and (A5). But, by (A4), c is a continuous function of k for k 2 (0; k), which tends to 0 as k tends to k; while, by (A5), k is a continuous function of c for c 2 (0; c) with k 2 ( k; k). Hence, there exists at least one value of k 2 ( k; k) that corresponds to a frictionless equilibrium.

If = 0, then we have f 0 ( k) = 0. If labor supply is inelastic, then, by (A1), c is a strictly decreasing function of k for all k 2 ( k; k), while, by (A3), c is a strictly increasing function of k for all k 2 ( k; k). Hence, the equilibrium value of k 2 ( k; k) must be unique. By continuity, the results also holds provided that is su¢ ciently small and that labor supply is su¢ ciently inelastic. 43 If the maximum feasible labor supply l is …nite, then, by (A1), c must tend to 0 while, by (A2), l must tend to l, as lim l! l v 0 (l) = 1. If l is in…nite, then c cannot tend to a strictly positive value, as, by (A2), l would also tend to a …nite value which would be inconsistent with (A1). Hence, c must tend to 0 while l must tend to 1.

44 Its derivative is equal to

45 Note that c cannot tend to 0 as lim K!0 0 (K) < 1.

by (36), which can be written as:

But, as tends to R , the …rst term of the product tends to in…nity while w tends to a strictly positive and …nite value. Hence, f (kL=l) must tend to zero while k and l also tend to strictly positive and …nite values. Thus, labor demand L must tend to zero as tends to R

. But, if L tends to zero while k and c tend to strictly positive and …nite values, then y s = Lf (k) must tend to zero while y d = c + kL tends to c > 0. Thus, in the limit, we unambiguously have y d > y s .

As tends to zero, R tends to R . Thus, c, k, w, and l for 2 R ; R converge to strictly positive and …nite values (since does not show up in (8)

while converges to R ). There is therefore a unique value of L that can equate aggregate supply y s = Lf (k) to demand y d = c + kL. But, for any > 0 and 2 R ; R , L is implicitly given by:

Thus, the unique value L that equates y s and y d determines a unique value of the ratio = R . Hence, for any strictly positive but arbitrarily small, there is a unique corresponding rate of in ‡ation 2 R

; R that is consistent with an equilibrium.

D Lemma 4

As in the proof of Lemma 3, let k = K=L, f (k) = F (k; 1), and

In a secular stagnation steady state with an exogenous real interest rate r, we must have

, by (16), and L [f (k) k] = c, by (24). 47 Output is given by y = c + kL, which implies that:

Hence, dy=dr < 0 if and only if " < " u .

E Proofs of Proposition 1, 2, 3, 4 and 5

As in the proof of Lemma 3, let k = K=L, f (k) = F (k; 1), and

We know that f (0) = 0, that f 0 (k) = kf 00 (k) > 0 for any k > 0, and that, for any K and L, F L (K; L) = f (K=L) K=Lf 0 (K=L) = f (K=L). Combining ( 15) and (36), we therefore have:

We also know, by (15), that w = f (k). The secular stagnation equilibrium (c; k; ; l; L; y) is therefore fully characterized by:

where I have added three new parameters, , %, and , that stand for the marginal utility of wealth, for the disutility of supplying labor, and for total factor productivity, respectively. To prove the propositions, we need to totally di¤erentiate these equations the …scal policy. We have:

where, to get the second line, I have used results from the proof of Proposition 4 with d =dg = 0. If (c + g) " u > c" , then, in the limit as tends to zero, an increase in g is welfare enhancing if and only if the term in the main square bracket is positive.

F Proof of Proposition 8

The steady state equilibrium (c; K; ; l; L) under the optimal policy with in ‡ation target ^ 2 R ; is fully characterized by:

where I have substituted the optimal wealth tax (78) into the demand for consumption (61) to obtain the …rst line and I have substituted the optimal corporate income tax policy (79) into the demand for investment resulting from (57) and (65) to obtain the second line. 51 When = ^ 2 R ; , the above equations that characterize the steady state 51 I have used the fact that, in steady state, the nominal interest rate must be equal to zero. This must be the case in the secular stagnation steady state as = ^ R (cf. footnote 14). The frictionless steady state is such that r = ^ , which implies that i = r + = ^ + = 0. equilibrium of the decentralized economy coincide with those that characterize the …rstbest allocation. Hence, there is a steady state equilibrium with = ^ that implements the …rst-best allocation (as we already know from Proposition 7). However, proceeding as in the proof of Lemma 3, it can be shown that a secular stagnation steady state equilibrium with 2 R

; R also exists provided that c > 0 when = R , i.e. provided that + R r n ^ > 0 or equivalently < + R + ^ , where I have used the fact that r n = . 52 Clearly, if ^ > R , then the secular stagnation equilibrium cannot coincide with the frictionless equilibrium.

Let us now prove that, conversely, the two equilibria can coincide when ^ = R . The frictionless and secular stagnation equilibria are characterized by the same set of equations, except for one exception: the frictionless equilibrium satis…es L = l, while the secular stagnation equilibrium satis…es R F L (K; L) = F L (K; l). Clearly, when ^ = R , the frictionless equilibrium with = ^ = R satis…es all the equations that characterize the secular stagnation equilibrium, including R F L (K; L) = F L (K; l). 53 Hence, when ^ = R , there is a secular stagnation equilibrium that coincides with the frictionless equilibrium. If the secular stagnation equilibrium is unique, then it must coincide with the frictionless equilibrium.

Proceeding as in the proof of Lemma 3, it can be shown that, if is su¢ ciently small, the secular stagnation steady state equilibrium is unique. 54 52 As > ^ , the optimal policy increases the range of values of for which a secular stagnation equilibrium exists. 53 The secular stagnation equilibrium has so far been de…ned for 2 R ; R , rather than 2

R

; R , since it typically satis…es L < l, which requires < R . However, it must fundamentally be characterized by the …rst four equations of this proof together with R F L (K; L) = F L (K; l). When the optimal policy with ^ = R is implemented, it turns out that the secular stagnation equilibrium is consistent with L = l and, hence, with = R . 54 Alternatively, uniqueness can be shown by proving that, for su¢ ciently small, any secular stagnation steady state must satisfy dy s =d > dy d =d , where y s = F (K; L) and y d = c + K.