Xavier Raurich

Thomas Seegmuller
email: thomas.seegmuller@univ-amu.fr.

On the interplay between speculative bubbles and productive investment

Keywords: JEL classi…cation: E22, E44, G12 Bubble, E¢ ciency, Vintage capital, Credit, Short sellers, Overlapping generations

The aim of this paper is to study the interplay between long term productive investments and more short term and liquid speculative ones. A three-period lived overlapping generations model allows us to make this distinction. Agents have a portfolio decision. When young, they can invest in human capital that is a productive long term investment that provides a return during the following two periods. When young or in the middle age, they can invest in a bubble. Young individuals can also borrow on a credit market to …nance the productive investment. However, the amount borrowed is limited by a credit constraint. We show that the existence of a stationary bubble raises productive investment and production when the bubleless economy is credit constrained and dynamically e¢ cient. Indeed, young agents sell short the bubble to increase productive investments, whereas traders at middle age transfer wealth to old age. The bubble allows to relax the credit constraint. We outline that a permanent technological shock inducing either a larger return of capital in the short term or a similar increase in the return of capital in both periods raises productive capital, production and the bubble size. We use our framework to discuss the e¤ect on the occurrence of bubbles of …nancial regulation and …scal policy.

Introduction

In recent years there is renewed interest in studying the link between productive and purely speculative investments. Some questions that naturally emerge are whether speculative investments are good or bad for capital accumulation and production, whether bubbles are compatible with dynamic e¢ ciency, and what is the role played by speculative assets. To address these issues, the literature focuses on the case where one may invest either in productive capital or in an asset without fundamental value, which is a pure bubble when its price is positive [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF], [START_REF] Weil | Con…dence and the real value of money in overlapping generations models[END_REF], [START_REF] Bosi | On rational exuberance[END_REF], [START_REF] Fahri | Bubbly liquidity[END_REF], [START_REF] Martin | Economic growth with bubbles[END_REF], [START_REF] Hirano | Asset bubbles, endogenous growth, and …nancial frictions[END_REF]). 1 In this literature, these two assets provide returns in the same period and are traded in each period. If we de…ne liquidity as the capacity to immediately trade an asset, both assets have the same liquidity.

The purpose of this paper is to study the existence of bubbles and their e¤ect on production when we introduce a di¤erential in the liquidity of assets. As in [START_REF] Diamond | Bank runs, deposit insurance and liquidity[END_REF], we consider two assets that provide returns in di¤erent periods. In particular, there are a speculative asset which gives returns in the short run and can be traded in each period and a productive asset which is only purchased when young and provides returns in the longer run. Obviously, this distinction implies that the speculative asset is more liquid than the productive one. A clear example of investment with returns in the long term is investment in human capital through education. This investment takes place during youth, implies returns during the life-cycle, while it depreciates with death. Accordingly, in this paper, we will identify productive capital with human capital. 2The model we examine is an overlapping generations model with three-period lived households. When young, households can borrow through debt and invest in two assets: human capital, which is used in production and give returns in middle and old ages, and an asset without fundamental value, which is traded also in middle age. This asset is a bubble when it is positively valued. In this case, we roll over this asset. We assume that the amount invested is limited by a credit constraint that uses a fraction of the returns from human capital in middle age as collateral. A smaller value of this fraction limits the amount of credit. In middle age, agents …nance credit and can invest only in the bubble to transfer purchasing power to old age. 3 It is important to underline that this framework introduces the notion that capital investment is less liquid than the speculative investment.

In accordance with the empirical evidence showing that bubbles occur in periods of large GDP growth [START_REF] Caballero | Speculative growth: hints from the U.S. economy[END_REF], Martin and Ventura (2012)), we show that bubbles may be productive because they are able to raise production through larger investments in human capital. Considering that the credit constraint is binding at the bubbleless steady state and that this steady state is dynamically e¢ cient, we show that the bubbly steady state features a higher level of production, because the bubble is used to …nance productive investment and relax the binding credit constraint. Therefore, our model is also consistent with the empirical evidence which highlights that credit is higher in periods of expansions [START_REF] Martin | Managing credit bubbles[END_REF]) and of bubbles (Jorda et al. (2015)).

The underlying mechanism explaining the former results is based on the interaction between two e¤ects, a crowding-in e¤ect through credit and a crowdingout e¤ect through saving. The …rst one corresponds to the fact that young agents sell short the speculative asset to …nance investment in productive human capital when the bubbleless steady state is dynamically e¢ cient and credit constrained. This allows to have a larger human capital and production than in the credit constrained bubbleless economy. Selling short the speculative asset can be an equilibrium because middle age agents reimburse the amount borrowed and also buy the speculative asset to transfer purchasing power to old age. This saving in middle age allows to have a bubble with …nite value in equilibrium and it corresponds to the crowding-out e¤ect, which allows to reach the golden rule and also to give a positive value to the bubble. Hence, starting with a dynamically e¢ cient bubbleless steady state, which is the most plausible assumption [START_REF] Abel | Assessing dynamic e¢ ciency: theory and evidence[END_REF]), the bubbly steady state attains the golden rule.

The bubble reallocates the income of middle age households to young and old ones. This means that the demographic structure of overlapping generations play a key role because, …rst, some agents are able to be short sellers of the bubble asset and second, there are heterogeneous investors in the market, young and middle age agents. 4 It is already known that heterogeneous investment opportunities play a key role to generate bubbles enhancing production [START_REF] Hirano | Asset bubbles, endogenous growth, and …nancial frictions[END_REF], [START_REF] Kocherlakota | Bursting bubbles: consequences and cures[END_REF]). We contribute to this literature by showing that a productive bubble also arises if it reallocates resources towards young agents, by generating liquidity used to raise human capital. 5Since the long term investment in human capital is a key element of our story, we study the e¤ect on the level of capital and bubble in the long-run of di¤erent types of permanent technological shocks that modify the short and long run returns of productive investment. We compare the steady states before and after the shock and we show that a biased technological shock implying a larger return in the longer term may reduce capital. This happens because the bubble may disappear and, hence, it can no more be used to …nance productive investment. On the contrary, if the technological shock is biased toward short term return of capital or increases the returns of capital in the short and long terms in like manner, we observe an increase in capital, production and bubble size. These shocks are typically the result of innovations. Then, our results explain that episodes of bubbles are associated with new innovations, as is documented and discussed in several contributions (see for instance [START_REF] Caballero | Speculative growth: hints from the U.S. economy[END_REF], [START_REF] Lansing | Speculative bubbles and overreaction to technological innovation[END_REF][START_REF] Lansing | Speculative growth, overreaction, and the welfare cost of technology-driven bubbles[END_REF], [START_REF] Scheinkman | Speculation, trading and bubbles[END_REF]).

We further apply our framework to the debate on the design of the …scal policy that aims to promote long term investment [START_REF] Wehinger | Fostering long-term investment and economic growth, summary of a high-level OECD …nancial roundtable[END_REF]). We show that the di¤erence between human capital and labor income taxes determines both the existence and the nature of the bubble, productive or unproductive. 6Moreover, we highlight that the suitable …scal policy to promote long term investment crucially depends on the existence and nature of bubbles.

Finally, to clarify the contribution of this paper, we recall that …rst results in the literature show that investment in a speculative bubble reduces capital accumulation in the long run to reach the golden rule [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF]). This requires that the steady state without bubble is dynamically ine¢ cient. 7 Dynamic ine¢ ciency has been criticized by [START_REF] Abel | Assessing dynamic e¢ ciency: theory and evidence[END_REF] for its lack of empirical relevance and, moreover, the lower level of production when there is a bubble is not observed in data, as it is for instance illustrated by [START_REF] Caballero | Speculative growth: hints from the U.S. economy[END_REF] and [START_REF] Martin | Economic growth with bubbles[END_REF]. Recently, several papers have provided answers to these two criticisms. First, Martin andVentura (2012, 2016) show the existence of productive bubbles when there are heterogeneous returns of capital investments and there is a process of bubble creation. Second, Fahri and Tirole (2012) and [START_REF] Hirano | Asset bubbles, endogenous growth, and …nancial frictions[END_REF] show the existence of productive bubbles when a …nancial constraint is introduced. In these papers, the bubble generates an investment multiplier e¤ect. Third, [START_REF] Kocherlakota | Bursting bubbles: consequences and cures[END_REF], and Miao and Wang (2011) also consider …nancial constraints that attribute to the bubble the role of collateral. As the bubbles allows to borrow larger amounts, it becomes productive. 8 Our paper contributes to this literature on the existence of productive bubbles. However, we di¤er from the …rst papers because our analysis does not need bubble shocks and from the last ones because our result precisely relies on the role of the bubble in relaxing a binding credit constraint. We show that the economy attains a larger aggregate production in the bubbly economy, when the bubbleless one is credit constrained and dynamically e¢cient. In contrast to [START_REF] Martin | Managing credit bubbles[END_REF], the crowding-in dominates the crowding out e¤ect when the bubble allows the credit constraint to be no more binding.

This paper is organized as follows. In the next section, we present our framework. In Section 3, we study the equilibria without and with bubbles to show the existence of productive bubbles. In Section 4, we discuss the e¤ect of technological shocks. Section 5 is devoted to study the e¤ect of …scal policy. Section 6 concludes, while many technical details are relegated to the Appendix.

Model

Time is discrete (t = 0; 1; :::; +1) and there are two types of agents, households and …rms.

Households

We consider an overlapping generations economy with constant population size. Each generation is populated by a continuum of mass one of agents that live for three periods. Each household has utility for consumption at each period of time. Preferences of an individual born in period t are represented by the following utility function:

u (c 1;t ) + u (c 2;t+1 ) + 2 u (c 3;t+2 ) (1)
where 2 (0; 1) is the subjective discount rate and c j;t amounts for consumption when young (j = 1), in middle age (j = 2), and when old (j = 3). For tractability, we assume that u(c j;t ) = ln c j;t . The household supplies one unit of labor when young. She shares her labor income, given by the wage w t , between consumption c 1;t and a portfolio of three assets: productive capital h t+1 ; debt d t and a speculative asset b 1;t . On the one hand, to capture the idea that, in contrast to investing in the liquid speculative asset, productive investment is less liquid and engages the household in the long term, we assume that capital provides a return 1 q t+1 in the second period of life and a return 2 q t+2 in the third period. We also assume that capital depreciates completely after being used in production. These properties of productive capital in this model are those of human capital: it is a long term investment done by young individuals that completely depreciates at the end of life. Accordingly, we interpret h t+1 as human capital. On the other hand, we assume that each young household may incur a debt d t that will be reimbursed in middle age R d t+1 d t : This loan is granted by middle age households living at the same period and it will be reimbursed when they are old.

The middle age household shares her income coming from human capital, the return on the speculative asset R t+1 and the payment of the debt accumulated when young between consumption c 2;t+1 ; the debt lent to young households d t+1 and a new investment in the speculative asset b 2;t+1 . Finally, when old, the household's consumption c 3;t+1 is equal to her income coming from the return of human capital, the reimbursement of debt made by middle age agents and the investment in the speculative asset done in middle age and remunerated at the return R t+2 . Accordingly, the budget constraints in each period of life of an agent born in period t are:

c 1;t + h t+1 + b 1;t = w t + d t (2)
c 2;t+1 + b 2;t+1 + d t+1 = 1 q t+1 h t+1 + R t+1 b 1;t R d t+1 d t (3)
c 3;t+2 = 2 q t+2 h t+1 + R t+2 b 2;t+1 + R d t+2 d t+1 (4)
As argued by [START_REF] Scheinkman | Speculation, trading and bubbles[END_REF], borrowers are often forced to cover their short positions with collateral. Following Hirano and Yanagawa (2017), Le [START_REF] Van | Intertemporal equilibrium with …nancial asset and physical capital[END_REF] and [START_REF] Martin | Managing credit bubbles[END_REF], credit and short sales on the speculative asset are limited by the following constraint:9

R d t+1 d t R t+1 b 1;t 6 1 q t+1 h t+1 (5)
with 2 [0; 1). This constraint limits the borrowing possibilities of young agents. In particular, we assume that the reimbursement in middle age of credit and short sales on the speculative asset is limited by the returns on human capital at this period. Hence, if this constraint is binding, human capital acts as a collateral. The parameter measures the borrowing limit. If = 0; credit and short sales are not possible. As increases, the constraint relaxes. We interpret as a policy parameter set through regulations. Note that (5) also implies that the income in middle age is strictly positive. Finally, we do not introduce such a constraint on the investment decision in middle age, because middle age agents do not borrow through credit and bubble short sales, meaning that the introduction of such a constraint is irrelevant. In the following, we call a constrained equilibrium (steady state), an equilibrium (steady state) where the constraint (5) is binding.

The household determines her optimal choices maximizing her utility (1) under the budget constraints (2)-( 4) and the credit constraint [START_REF] Boucekkine | Obsolescence and modernization in the growth process[END_REF]. By direct inspection of her program and by lack of arbitrage opportunity, we immediately deduce that R d t+1 = R t+1 for all t. We can then de…ne x 1;t = b 1;t d t and x 2;t = b 2;t + d t ; and rewrite the constraints (2)-(5) as follows:

c 1;t + h t+1 + x 1;t = w t (6) c 2;t+1 + x 2;t+1 = 1 q t+1 h t+1 + R t+1 x 1;t (7) c 3;t+2 = 2 q t+2 h t+1 + R t+2 x 2;t+1 (8) R t+1 x 1;t > 1 q t+1 h t+1 (9)
At this point, it is important to clarify the interpretation of the returns of the two assets: human capital and the speculative asset. On the one hand, the returns of human capital obtained in the second and third periods of life can be interpreted as the wage of skilled workers, the skill being acquired during youth. The wage w t obtained in the …rst period of life will be simply the labor compensation of unskilled young workers. Moreover, since human capital investment provides returns in the following two periods, this introduces the di¤erence between short term speculative investments and long term productive investments, the …rst ones being more liquid than the second ones, which is the key ingredient of our story.

On the other hand, we clarify that the speculative asset is supplied in one unit at a price p t at period t. New investments in this asset by young and middle age agents are in quantities t and 1 t , respectively. Therefore, the values of this asset bought or sold by these agents are b 1;t = p t t and b 2;t = p t (1 t ). Of course, if either b 1;t < 0 or b 2;t < 0, one type of trader is a short seller of this asset. It corresponds to equilibria where either t < 0 or t > 1. Finally, since this asset has no fundamental value, it is a bubble if p t = b 1;t + b 2;t > 0, whereas there is no bubble if p t = b 1;t + b 2;t = 0 and b 1;t = b 2;t = 0. Taking into account the de…nition of the variables b 1;t and b 2;t , we obtain:

b 1;t+1 + b 2;t+1 = R t+1 (b 1;t + b 2;t ) (10)
where the left-hand side of (10) measures the value of the bubble net purchasing in period t + 1 and the right-hand side the value of the net sales in that period. It follows that the return R t+1 is the growth factor of the bubble price. Finally, using the de…nitions of x 1;t and x 2;t ; equation (10) rewrites as:

x 1;t+1 + x 2;t+1 = R t+1 (x 1;t + x 2;t ) (11)
The variable x 1;t measures the value of the short term assets used to smooth consumption between the …rst two periods of life, whereas x 2;t measures the value of the short term assets used to smooth consumption between the last two periods. When these variables take a positive value, individuals use the short term assets to postpone consumption and they use these assets to borrow when they take a negative value. To clarify the interpretation of short term assets, it is important to distinguish between two di¤erent cases. First, if x 1;t + x 2;t = 0, x 1;t = d t and x 2;t = d t , then the short term asset is simply credit. In contrast, if x 1;t + x 2;t > 0, short term assets involve a bubble.

An equilibrium where x 1;t + x 2;t > 0, i.e. b 1;t + b 2;t > 0, can alternatively be interpreted as a situation with a bubble on credit. With this interpretation of our short term assets, there is a market for assets that value credit. When young individuals obtain a credit from a …nancial institution, this institution can sell this asset to another …nancial institution that buys it using the deposits of middle age individuals. When the demand of this asset is larger than the supply of credit, there is a bubble. Such a situation clearly corresponds to our scheme where young individuals sell (short) a speculative asset to middle age individuals.

Households maximize the utility (1) subject to the budget constraints ( 6)-( 8) and the credit constraint [START_REF] Fahri | Bubbly liquidity[END_REF]. 10 The solution of the household's problem when the credit constraint is binding is given by the following equations:

x 1;t = 1 q t+1 R t+1 h t+1 (12)
c 3;t+2 = R t+2 c 2;t+1 (13)
c 2;t+1 1 1 q t+1 R t+1 = (1 ) 1 q t+1 + 2 q t+2 R t+2 c 1;t (14)
and the constraint is binding if c 2;t+1 > R t+1 c 1;t . On the contrary, if the credit constraint is not binding, the solution of the household's problem is given by the following equations:

c 2;t+1 = R t+1 c 1;t (15) c 3;t+2 = R t+2 c 2;t+1 (16)
1 = 1 q t+1 R t+1 + 2 q t+2 R t+1 R t+2 (17)

Firms

Firms produce with the following technology:

Y t = H t L 1 ; with 2 (0; 1=2)
where L is the number of young workers and H t is aggregate productive capital composed by the stock of human capital of generations born at period t 1 and t 2. Since households, that live three periods, invest in human capital when young and this investment has returns in middle and old ages, human capital is two-period lived, and completely depreciates after. We assume that the di¤erent human capitals are perfect substitutes in the production

H t = 1 h t + 2 h t 1 (18)
where 1 > 0 and 2 > 0 measure the productivities of the human capital of the generations born in t 1 and t 2; respectively. 11 In a sense, h t corresponds to vintage capital of the One-Hoss Shay type with a lifetime of two periods, but with di¤erent productivities during the lifetime (see [START_REF] Benhabib | Vintage capital, investment and growth[END_REF] and Boucekkine et al. (2005a)). Since L = 1, pro…t maximization under perfect competition implies that the wage w t and the return q t from aggregate human capital H t are given by:

w t = (1 )H t ( 19
)
q t = H 1 t ( 20
)
1 1 We do not allow to have 2 = 0. Indeed, in this case, capital is no more a long term asset.

Existence of productive bubbles

We recall that a bubble is productive if the levels of human capital and production at a steady state with bubble are larger than the levels attained at a steady state without bubble. Our purpose is to show that the speculative bubble can be productive when it relaxes the credit constraint and promotes investment in human capital. Therefore, we will analyze and compare the credit constrained bubbleless steady state and the unconstrained bubbly steady state. In Appendix B.1, we characterize the constrained bubbly steady state and show that the bubble cannot be productive in this case. Hence, the bubble is productive when it facilitates the access to credit, by making the credit constraint unbinding.

Constrained bubbleless steady state

We …rst analyze the model without bubble, i.e. b 1;t = b 2;t = 0, and with the credit constraint ( 9) binding. In this case, x 2;t = x 1;t : Using ( 12), the budget constraints ( 6)-( 8) rewrite as:

c 1;t = w t h t+1 1 1 q t+1 R t+1 (21)
c 2;t+1 = (1 ) 1 q t+1 h t+1 1 q t+2 h t+2 R t+2 (22)
c 3;t+2 = q t+2 ( 1 h t+2 + 2 h t+1 ) (23)
Substituting ( 21)-( 23) into ( 13) and ( 14), we get:

q t+2 [ 1 (1 + )h t+2 + 2 h t+1 ] = R t+2 (1 ) 1 q t+1 h t+1 ( 24
)
1 1 q t+1 R t+1 = h 1 q t+1 (1 ) + 2 qt+2 Rt+2 i h w t h t+1 1 1 qt+1 Rt+1 i (1 ) 1 q t+1 h t+1 1 qt+2ht+2 Rt+2 (25)
and, using [START_REF] Kamihigashi | The spirit of capitalism, stock market bubbles and output ‡uctuations[END_REF], the credit constraint is binding if:

(1

) 1 q t+1 + 2 q t+2 R t+2 > R t+1 1 1 q t+1 R t+1 (26)
De…nition 1 Given h 1 > 0 and h 0 > 0; a credit constrained equilibrium without bubble is a path of fh t ; q t ; R t ; w t ; H t g 1 t=0 that solves equations ( 18), ( 19), ( 20), [START_REF] Miao | Introduction to economic theory of bubbles[END_REF], and ( 25), and satis…es (26). 12 At an interior steady state, we have w=h = q( 1 + 2 )(1 )= . Using equations ( 18), ( 20), ( 24) and ( 25), we deduce that:

R = 1 (1 + ) + 2 (1 ) 1 R (27)
q = 1 (1+ ) 1+ + 2 ( 1 + 2 ) 1 + 2 1 (1 ) 1 (1+ )+ 2 q (28) h = q 1 1 1 1 + 2 h ( 29
)
At this steady state the credit constraint must be binding, which requires c 2 > Rc 1 . Using [START_REF] Samuelson | An exact consumption-loan model of interest with or without the social contrivance of money[END_REF], this is satis…ed at a steady state if R < 1 and:

1 q(1 ) + 2 q=R 1 1 q R > 1
The …rst inequality is satis…ed for

< 1 2 1 (1 + 2 ) and 2 = 1 < (30)
Since R > 1 q, the second inequality is equivalent to

1 1 (1 + ) + 2 1 + 2 1 + + 2 1 + 2 1 1 > : (31)
The left-hand side is decreasing in and strictly smaller than 1, whereas the right-hand side is of course linearly increasing in . Hence, there exist a e < 1 such that the previous inequality holds if < e and

1 > 1 + 2 1 1 + + 2 (32)
The previous arguments are summarized in the following proposition:

Proposition 1 There is a unique constrained bubbleless steady state h if:

1. < minf ; e g; 2. 2 = 1 < ; 3. 1 > 1 + 2 1 1+ + 2 .
At a steady state, the resource constraint on the goods market writes as c 1 +c 2 +c 3 = [( 1 + 2 )h] h, where the left-hand side of the resource constraint is total consumption at the steady state and the right-hand side is production net of investment. The steady state is dynamically e¢ cient if it is not possible to increase total consumption at one date without decreasing total consumption at another date. De la Croix and Michel (2002, pages 82-90) show that the steady state is dynamically e¢ cient when there is underaccumulation of capital, which occurs when net production increases with capital investment. They also show that an equilibrium is Pareto optimal when there is under-accumulation, meaning that dynamic e¢ ciency where total consumption is maximized also implies lifetime utility maximization. It can be shown that this occurs when the net return of investment is positive, i.e. ( 1 + 2 )q > 1. 13 Using equation [START_REF] Shiller | Irrational Exuberance[END_REF], this is satis…ed under the following condition: 1)

1 > (1 + ) 1 + + 2 + 1 1 1 + 2 1 (
1 (1 + ) + 2 (33)
The right hand side of the dynamic e¢ ciency condition is the saving rate for human capital investment over the wage h=w. Therefore, the dynamic e¢ ciency condition states that this saving rate must be smaller than = (1 ).

Unconstrained bubbly steady state

We study now the unconstrained equilibrium with a positive bubble, i.e. x 1;t + x 2;t > 0: Therefore, we use the …rst order conditions of the consumers'problem when the credit constraint is not binding and the budget constraints to obtain the following de…nition.

De…nition 2 Given h 1 > 0 and h 0 > 0; an equilibrium with bubbles is a path of fh t ; H t ; R t ; q t ; w t ; c 1;t ; c 2;t ; c 3;t ; x 1;t ; x 2;t g1 t=0 that satis…es the budget constraints ( 6), ( 7) and ( 8), the market clearing conditions [START_REF] Hirano | Asset bubbles, endogenous growth, and …nancial frictions[END_REF] and [START_REF] Lagos | A uni…ed framework for monetary theory and policy analysis[END_REF], the competitive factor payments [START_REF] Lansing | Speculative bubbles and overreaction to technological innovation[END_REF] and [START_REF] Lansing | Speculative growth, overreaction, and the welfare cost of technology-driven bubbles[END_REF], and the …rst order conditions (15), ( 16) and [START_REF] Lagos | Asset prices and liquidity in an exchange economy[END_REF].

We proceed to obtain the steady state of this equilibrium. From [START_REF] Hirano | Asset bubbles, endogenous growth, and …nancial frictions[END_REF], a bubbly steady state is such that R = 1. From ( 17) and [START_REF] Lansing | Speculative growth, overreaction, and the welfare cost of technology-driven bubbles[END_REF], we obtain:

q = 1= ( 1 + 2 ) q (34) h = 1 1 ( 1 + 2 ) 1 h (35)
Note that (34) implies that the bubbly steady state maximizes the production net of investment and, hence, this bubbly steady state attains the golden rule. Indeed, the golden rule is de…ned as the steady state stock of human capital that maximizes total consumption at a stationary path and, hence, it is the stock of human capital that maximizes production net of investment. Note that this implies that any regulation or …scal policy will reduce welfare in this economy. It also implies that the level of welfare at a steady state with bubbles is larger than the level of welfare attained at a steady state without bubbles.

Next, we use ( 15) and ( 16) to obtain c 2 = c 1 and c 3 = 2 c 1 . We substitute these two relationships and w=h = q 6)-( 8) to obtain:

( 1 + 2 ) (1 )= = (1 )= in (
x 2 = 2 1 + + 2 2 2 + 1 1 w x 2
(36)

x 1 = (1 + ) 1 + + 2 1 w x 1 (37) x 1 + x 2 = + 2 2 1 + + 2 2 2 + 1 2 + 1 1 w ( 38
)
where w = (1 )h= is the equilibrium wage. The existence of this steady state depends on the bubble being positive, x 1 + x 2 > 0; and the constraint being non-binding. While the …rst condition immediately follows from (38), the second condition follows from assuming that the constraint (9) holds in strict inequality in this steady state, i.e. x 1 > 1 qh: Using (34), ( 35) and (37), the constraint is non-binding when:

> 1 + 2 1 1 1 (1 + ) 1 + + 2 : ( 39
)
The following proposition establishes the existence of an unconstrained bubbly steady state: Proposition 2 There is a unique bubbly steady state h if and only if:

1. > ; 2. 1 < +2 2 1+ + 2 2 + 1 2 2 + 1 .

Productive bubbles

We proceed to prove the existence of productive bubbles by showing that the equilibrium stock of human capital is larger at the bubbly steady state than at the bubbleless one. However, we …rst must guarantee that both steady states may coexist.

Proposition 3

The unconstrained bubbly steady state (h) and the constrained bubbleless steady state (h) coexist if the following conditions are satis…ed:

1. 2 1 < ; 2. < < e ; 3. 1 + 2 1 1+ + 2 < 1 < (1+2 ) 1+ + 2 1 + 2 1 +2 2 . Proof. See Appendix A.2.
Recall that the parameter determines the strength of the credit constraints and can be interpreted as a policy parameter that measures the degree of regulation in the …nancial sector. The larger is, the more deregulated is the …nancial sector, since a young trader is less constrained to increase her borrowing. Therefore, the value of this parameter crucially determines the existence of the two steady states. On the one hand, if is low, < ; the constraint becomes binding in the bubbly equilibrium. On the other hand, if is too large, > e ; the credit constraint in the bubbleless equilibrium may be non-binding. Therefore, the coexistence of the two steady states occurs for intermediate values of ; 2

; e : This result can be connected to some …ndings of Fahri and Tirole (2012), who rather interpret as the degree of pledgeability. In contrast to them, a bubble may exist if the degree of pledgeability is high enough, because it may exists when the credit constraint is no more binding. Our result is also related to Hirano and Yanagawa (2017), who show in a model with heterogeneous productive investments that an intermediate degree of pledgeability is required for the existence of a bubble.

The bubble is productive if and only if h > h. This is equivalent to have q < q. Using ( 28) and (34), this means:

F ( ) (1 ) (1 + ) + 2 = 1 < 2 1 + 1 1 1 (1 + ) 1 + + 2 (40)
Note that the condition (40) holds if and only if condition (33) is satis…ed. Therefore, the bubble is productive only when the bubbleless steady state is dynamically e¢ cient and, hence, the bubble is used to raise investment. Condition (40) requires 1 > (1+ ) 1+ (1+ ) , which implies > 0. It can be shown

that 1 + 2 1 1+ + 2 < (1+ ) 1+ (1+ ) < (1+2 ) 1+ + 2 1 + 2 1 +2 2 under 2 = 1 <
. Therefore, we focus on parameter con…gurations that satisfy:

(1 + ) 1 + (1 + ) < 1 < (1 + 2 ) 1 + + 2 1 + 2 1 + 2 2 :
(41)

Our main result follows: We observe that the bubble is productive only when 1 > (1+ ) 1+ (1+ ) ; which implies that x 1 < 0. Since x 1 + x 2 > 0, it means that x 2 > 0. On the one hand, x 1 < 0 implies that the bubble allows young agents to invest more in human capital by having a short position on the speculative asset. As a consequence, the bubble is bene…cial for output, since output is larger than in the bubbleless steady state. It is the credit role of the bubble and corresponds to a crowdingin e¤ect. It relaxes the credit constraint because borrowing through credit and short sale of the bubble is used by young individuals to increase investment in human capital.

Proposition 4 If 2 1 <
On the other hand, x 2 > 0 implies that middle age households …nance the amount borrowed by the young and also generate the positive value of the bubble. This induces a crowding-out e¤ect that fosters the economy to reach the golden rule, whose human capital stock is h. This last e¤ect, already identi…ed in the seminal paper by [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF], has a negative impact on production but improves consumption since it allows to reach the golden rule. In fact, both the credit and the crowding-out e¤ects imply a reallocation from middle age to young and old ages, since at middle age the household reimburses the amount borrowed when young and invests in the bubble to raise consumption when old. However, the bubble is productive when the crowding-in e¤ect dominates the crowding-out one. This happens when is high enough, because the credit constraint is more easily relaxed, following an increase of human capital.

The demographic structure of overlapping generations play a key role because it introduces heterogeneous traders in the market, young and middle age agents. This heterogeneity explains that the bubble increases production. As young agents are short sellers of the speculative asset, they obtain the liquidity needed to raise productive investment.

It is worth mentioning that if 1 < (1+ ) 1+ (1+ ) , we have x 1 > 0. Young agents do not borrow to raise human capital and production. In this case, condition (40) is not satis…ed and we have q > q and h < h. In other words, the existence of the bubble reduces human capital and production. In fact, when

1 < (1+ )
1+ (1+ ) the credit constrained equilibrium without bubbles is ine¢ cient and the only role played by the bubble is to crowd-out capital, as highlighted in [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF].

In this paper, we identify a new mechanism, based on the short sale of the bubble and the relaxed credit constraint, through which the bubble raises production, which is in accordance with the empirical evidence. This also means that if the bubble suddenly crashes, production will reduce, which is also in accordance with what we observe. These di¤erent facts are well documented by [START_REF] Caballero | Speculative growth: hints from the U.S. economy[END_REF] and [START_REF] Martin | Economic growth with bubbles[END_REF], among others.

As we have already mentioned, the parameter can be seen as a policy parameter that measures the …nancial regulation. The larger is, the more deregulated is the economy since a young trader is allowed to borrow more. To analyze the role of deregulation in our framework, we study more accurately the e¤ect of the policy parameter on human capital. To this end, …rst note that h does not depend on (see equation ( 35)), whereas h varies with . 14 Starting with a low value of ; an increase in this parameter has a direct e¤ect through the credit constraint, increasing borrowing and investment. However, the interest rate R also increases, which deters borrowing and investment in human capital. Eventually, this second e¤ect dominates and h decreases with for a su¢ ciently large : This explains that when increases from to e , we have …rst h > h. The di¤erence between these two capital stocks …rst increases and, eventually, decreases. When crosses b , h = h, and h decreases and becomes lower than h for all 2 b ; e . Therefore, deregulation makes the bubble productive and, in fact, increases the di¤erence between the two steady state stocks of human capital. However, more deregulation also means a larger loss of capital in the case of a crash of the bubble.

Technological shocks

Since illiquid human capital is a key ingredient of our results, we study in this section the e¤ects on both the bubbly and bubbleless steady states of permanent technological shocks that a¤ect the liquidity of human capital. These technological shocks consist on modifying the productivities 1 or 2 . In this way, we can study the e¤ect of a technological shock biased toward either short term return ( 1 relative to 2 increases) or a long term return ( 2 relative to 1 increases). Increasing 1 and 2 proportionally, we also investigate the case of a technological shock neutral with respect to the short versus long term returns of capital. We assume that we are in the most interesting con…guration, where the constrained bubbleless and the unconstrained bubbly steady states coexist and the bubble is productive, and we analyze the e¤ect of permanent technological shocks on long run human capital accumulation. We consider …rst the constrained bubbleless steady state.

1+ + 2 1 + 2 1 +2 2
, human capital h increases following an increase of either 1 or 1 and 2 in the same proportion. When 2 increases, the e¤ ect on h is undetermined.

Proof. See Appendix A.4. 1 4 To see this, we combine ( 29) and (40), to obtain:

h = 1 1 1 + 2 (1 + ) 1 + + 2 ( 1 + 2 ) 1 + 1 F ( ) 1 1
An increase of 1 pushes up productive investment when young for two main reasons. First, it raises the return of human capital in middle age. Second, in the credit constrained steady state, it allows to borrow more when young to …nance human capital investment. In contrast, an increase of 2 has two opposite e¤ects on human capital investment. A positive one because it raises the return of investment in old age. However, this increase of the return from human capital in old age incites a middle age household to save less, i.e. to …nance less credit through a lower x 2 . In the equilibrium without bubbles, the reduction of x 2 directly reduces borrowing (x 1 less negative) and, therefore, decreases investment in human capital. Obviously, the interaction between these two opposite e¤ects explains the ambiguous e¤ect of 2 on human capital accumulation.

We now focus on the unconstrained steady state with a positive bubble:

Proposition 6 Assuming that 2 1
< , b < < e and 1 strictly lower but su¢ ciently close to (1+2 )

1+ + 2 1 + 2 1 +2 2
, we have the following:

1. Productive investment h increases, while the return q decreases if there is an increase of either 1 , or 2 or of both, 1 and 2 ; in the same proportion;

2. x 1 decreases if there is an increase of either 1 , or 2 or of both, 1 and Proof. See Appendix A.5.

An increase of 1 increases the incentive to invest in human capital when young. As a consequence, the investment in human capital and, hence, the amount borrowed by young households increases (lower x 1 ). The middle age household has a larger income, even relatively to what she expects to have when old. Hence, she transfers a larger amount to old age, increasing her holding of the bubble x 2 . Despite the more signi…cant short position of the young on the speculative asset, the value of the bubble, x 1 + x 2 ; increases.

Following a raise of 2 , the di¤erence between the returns of human capital and the bubble when old increases. Therefore, there is a substitution e¤ect that pushes up productive investment, but pushes down the purchasing of the bubble x 2 in middle age. To increase productive investment, young agents borrow more (lower x 1 ). As both x 1 and x 2 decrease, the value of the bubble x 1 +x 2 decreases.

When there is a proportional increase of 1 and 2 , i.e. keeping 2 = 1 constant, the technological shock is neutral. Of course, capital investment raises because its return through the life-cycle is greater. This increase is …nanced by increasing borrowing through the speculative asset (lower x 1 ). Because the wage and the return from human capital in middle and old ages increase, there is no reallocation e¤ect in the portfolio choices, but more signi…cant positions of bubble holdings, meaning a higher x 2 . The value of the bubble x 1 +x 2 increases, because it linearly depends on income.

In Proposition 6, we have enlightened how the bubbly steady state evolves according to various technological shocks. We have shown that while human capital increases with all the technological shocks considered, the value of the bubble may either increase or decrease, depending on the technological shock being biased toward long term, or rather toward short term or neutral. In addition, by direct inspection of Proposition 2, we see that the relative return of investment between the long and short terms 2 = 1 a¤ects the existence of the bubbly steady state and, therefore, its coexistence with the bubbleless one. We study now more deeply what is the e¤ect of the di¤erent technological shocks on the existence of the bubbly steady state. This analysis is particularly important because we have assumed that 1 is strictly lower but su¢ ciently close to

(1+2 ) 1+ + 2 1 + 2 1 +2 2
; which is the condition to have a positive bubble and, hence, small technological shocks may a¤ect the existence of a bubble. To perform this analysis, note that x 1 + x 2 > 0 is equivalent to 2 = 1 < , with:

+2 2 1+ + 2 1 2 1 +2 2 1+ + 2 (42)
Hence, is strictly positive if:

Assumption 1 (1+ ) 1+ + 2 < 1 < +2 2 1+ + 2 .
We can also show that under 1 > (1+ ) 1+ (1+ ) , < . We then deduce the following proposition: Proposition 7 Assuming 2 = 1 < , b < < e and that Assumption 1 holds, the bubbly steady state coexists with the bubbleless steady state for all 2 = 1 < , the bubble collapses for 2 = 1 = , and there is only the bubbleless steady state for 2 = 1 > . This means that:

1. Any permanent neutral technological shock that increases both 1 and 2 proportionally does not a¤ ect the existence of the bubbly steady state;

2. A permanent technological shock biased toward short term return that increases 1 making 1 > 2 = allows the existence of the bubbly steady state;

3. A permanent technological shock biased toward long term returns that increases 2 making 2 > 1 rules out the existence of the bubbly steady state.

There is a level of 2 = 1 above which the bubbly steady state does no more exist and the only steady state is the bubbleless one. In contrast, for 2 = 1 low enough, there is a multiplicity of steady states, the bubbleless one coexists with the bubbly one. Indeed, if 2 is low regarding 1 , h is more a short term investment because its return is larger in the …rst period than in the second one. In this case, middle age agents transfer wealth to the future by using the speculative asset. This explains the existence of the bubble, which allows to have a steady state where human capital investment bene…ts from the bubble liquidities. On the contrary, if 2 is large regarding 1 , h is more a long term investment because its return is larger in the second period. In this case, middle age agents do not use the bubble to transfer wealth to the future and there is no bubble liquidities to raise human capital investment. Therefore, technological shocks that raise the ratio 2 = 1 may make the bubble crash.

Propositions 5-7 allow to have a general picture of long-run human capital following a permanent technological shock, biased toward short or long term returns, or which is neutral. Figure 1 illustrates the di¤erent cases. 15 Panel (a) shows that if agents start by coordinating their expectations on the bubbly steady state and there is a technological shock that permanently raises 2 , productive investment h increases if 2 is low enough. Then, there is a level of 2 above which the bubbly steady state disappears and productive investment may converge to the bubbleless steady state characterized by a lower level of h. When 2 crosses the value such that the bubbly steady state disappears, we may observe a decrease of productive investment h even if the technological shock increases its return in the long term. This means that a larger return in the long term does not necessarily imply higher levels of human capital. In this paper, the bubble provides liquidities to …nance human capital, but a too signi…cant return in the long-term rules out the bubble, and is therefore damaging for human capital itself.

Panel (b) of Figure 1 illustrates the opposite con…guration, which happens when there is a technological shock that raises 1 . There is a threshold level above which the bubbly steady state exists. Then, if agents coordinate their expectations on the bubbly steady state, any further raise of the return 1 implies a raise of capital. In addition, Proposition 7 allows us to argue that the bubble also enlarges under this type of technological shock.

Finally, if 2 = 1 < , a positive neutral technological shock has a similar e¤ect on both steady states. This is shown in Panel (c) of Figure 1. If agents coordinate their expectations on the bubbly steady state, such a technological shock induces a raise of capital and of the bubble size.

The last two results are especially interesting because they illustrate that in our model a permanent technological shock biased toward the short term or neutral between the short and the long terms is associated to larger levels of human capital, production and bubble size. These permanent positive shocks can be directly interpreted as innovations. Interestingly, there are many evidences showing that episodes of bubbles are often associated to new innovations (see for instance [START_REF] Caballero | Speculative growth: hints from the U.S. economy[END_REF], [START_REF] Lansing | Speculative bubbles and overreaction to technological innovation[END_REF][START_REF] Lansing | Speculative growth, overreaction, and the welfare cost of technology-driven bubbles[END_REF] and Scheinkman (2013)). 16 In the framework of this model, where human capital is formed by two groups of workers di¤erentiated by age, and 2 measures the productivity of the oldest group of workers, human capital obsolescence can be interpreted as a reduction of 2 relative to 1 . Therefore, innovations may facilitate the emergence of bubbles according to the results in Proposition 7. As a consequence, this paper contributes to this literature by providing a rationale on the observed relation between periods of large innovations and episodes of bubbles. Finally, this result may also be related to Hirano and Yanagawa (2017) who consider a model with heterogenous investment projects. In their framework, a technological progress in favor of the most productive investment promotes the occurrence of bubbles.

Technological shocks that make the bubble disappear will govern dynamic paths that converge to the constrained bubbleless steady state with lower capital. We are not able to address analytically the analysis of dynamic transitions. However, they can be studied numerically. This is done in Figure 2, where we show the transitional dynamics due to a technological shock that permanently increases the ratio 2 = 1 . 17 The …gure displays the transitional dynamics of by roads and highways, commercial radio broadcast and widespread electri…cation of manufacturing; (iii) In 1950s and 1960s, television, suburban life style and space travel; (iv) In late 1990s, internet, innovations in computers and information technology, and emergence of the web-based business model. 1 7 The value of the parameters in this numerical example are the following: = 0:4275; = 0:9; and = 0:164: Regarding the technological parameters, before the shock they are an economy that is initially in an unconstrained bubbly steady state and, after the shock that occurs in period 3, converges to a constraint bubbleless steady state. Panel (a) shows that following the shock the amount borrowed by young individuals to …nance human capital investment decreases (larger x 1 ). This is a consequence that, after the crash of the bubble, the economy is credit constrained, which limits the amount borrowed and, hence, reduces human capital investment. In fact, Panel (b) shows that human capital decreases during the transition. Panel (c) shows that q increases, which is a direct consequence of the reduction in human capital and Panel (d) shows that the return from debt, R t+1 ; is smaller than one along the entire transition in the credit constrained economy.

Fiscal policy

There is a debate on the design of the …scal policy that aims to promote long term investments [START_REF] Wehinger | Fostering long-term investment and economic growth, summary of a high-level OECD …nancial roundtable[END_REF]). The purpose of this section is to contribute to this debate by studying the e¤ect of taxes on both the existence of bubbles and on the stationary values of human capital. To this end, we consider a government that collects taxes on labor income, w < 1, and on human capital income, h < 1. As we interpret productive capital as human capital, h must be interpreted as the tax rate on large wages, as large wages are typically obtained by high skilled workers. In this context, the labor income tax, w ; must be interpreted as a tax on the wage of unskilled workers. Obviously, in a progressive tax system, w < h : We do not introduce a tax on the return of the speculative asset, nor on the return of the credit. On the one hand, the net return of the speculative asset is zero at a bubbly steady state and, thus, such a tax would be they guarantee that the bubbly equilibrium is unconstrained, and the bubbeless equilibrium is credit constrained along the entire transition. irrelevant in the long run. On the other hand, we study steady states without bubbles that are credit constrained. As we have seen before, in this steady state, the net return of the credit is negative because R < 1 and, hence, the tax on the return of the credit is also irrelevant.

We assume that government revenues are employed to …nance non-productive government spending G t . In this way, government spending does not distort individuals decisions. At each period, the government budget is balanced:

G t = w w t + h q t ( 1 h t + 2 h t 1 )
Note that if either w < 0 or h < 0, there is a subsidy on either labor or human capital. Such a policy scheme is in accordance with the government budget constraint, as long as G t > 0.

The introduction of this …scal policy modi…es the individual budget constraints as follows:

c 1;t + h t+1 + x 1;t = (1 w ) w t (43) c 2;t+1 + x 2;t+1 = (1 h ) 1 q t+1 h t+1 + R t+1 x 1;t (44) c 3;t+2 = (1 h ) 2 q t+2 h t+1 + R t+2 x 2;t+1 (45)
and the credit constraint becomes

R t+1 x 1;t (1 h ) q t+1 1 h t+1 (46)
We study now how taxes modify the long run equilibrium with and without bubble. We examine in particular which type of …scal policy promotes human capital accumulation.

To prove the existence of both the constrained bubbleless and the unconstrained bubbly steady states and compare them, we use the same methodology than in Section 3.

Constrained bubbleless steady state with taxation

Without bubble (x 1;t + x 2;t = 0), the solution of the consumer's problem when the utility function is (1), the budget constraints are (43), ( 44) and (45) and the credit constraint (46) is binding is characterized by the following …rst order conditions:

c 3;t+2 = R t+2 c 2;t+1 (47)
c 2;t+1 1 1 h 1 q t+1 R t+1 = c 1;t 1 q t+1 (1 ) + 2 q t+2 R t+2 (48)
and

c 2;t+1 > R t+1 c 1;t (49)
We take into account that in the bubbleless steady state x 2 = x 1 and we substitute (44)-( 46) in (47) to obtain the steady state return from the credit:

R = 2 + (1 + ) 1 (1 ) 1 R T
We proceed to obtain the steady state value of q: We substitute (43), ( 44) and ( 46) in (48) and use w = ( 1 + 2 ) qh (1 ) = to obtain:

1 1 + R R 1 1 1 h 1 q R = (1 w ) (1 h ) 1 ( 1 + 2 ) q 1 1 h + q 1 R 1 (1 ) + 2 R
Using the steady state interest rate, we get:

q = 1 (1 w ) 1 ( 1 + 2 ) (1+ ) 1+ + 2 + (1 h ) 1 (1 ) 1 2 + (1+ ) 1 q T (50)
and using ( 18)-( 20), we deduce that:

h = q T 1 1 1 1 + 2 h T w = (1 ) q T 1 w T
Note that both taxes increase q and reduce human capital. On the one hand, the tax on labor income reduces the net income available to invest in human capital. On the other hand, the tax on human capital reduces the collateral and, hence, it reduces investment in human capital as the credit constraint is binding.

Finally, this steady state is constrained if c 2 > Rc 1 , which occurs when R T < 1, i.e. < and 2 = 1 < , and:

1 1 (1 + ) + 2 1 + 2 1 + + 2 1 + 2 1 1 > (51)
where

1 h 1
w measures the asymmetric e¤ect of taxes on the di¤erent sources of income. The left-hand side is decreasing in and strictly smaller than 1, whereas the right-hand side is of course linearly increasing in . Hence, there exist a e T < 1 such that the previous inequality holds if < e T and

1 > 1 + 2 1 1 + + 2 (52)
Proposition 8 There is a unique constrained bubbleless steady state with taxation h T if:

1. < minf ; e T g;

2. 2 = 1 < ; 3. 1 > 1 + 2 1 1+ + 2 .
Finally, we note that > e T if and only if

1 < 1 + 2 2 2 + 1 + 2 2 1 + + 2 (53)
Note that the former proposition is a direct extension of Proposition 1, modi-…ed to include the taxes. It shows that …scal policy has no e¤ect on the existence of the constrained bubbleless steady state if w = h : Thus, only through asymmetric taxation of the two sources of income, …scal policy can a¤ect the existence of this equilibrium. In particular, this steady state exists when is large and, hence, either h is small or w is large. When h is small, the return after taxes of investment is large and, thus, individuals borrow until the constraint binds to increase investment. When w is large, the wage after taxes is small, which limits investment in human capital. Individuals are forced to borrow until the constraint binds to invest in human capital.

Unconstrained bubbly steady state with taxation

Maximizing the utility (1) under the budget constraints (43)-( 45), we show that, at the bubbly equilibrium (x 1;t + x 2;t > 0), the consumer's optimal choices satisfy:

R t+1 = (1 h ) q t+1 1 + q t+2 2 R t+2 c 2;t+1 = c 1;t R t+1 c 3;t+2 = c 2;t+1 R t+2
At the bubbly steady state, R = 1: Then, the previous equations simplify to c 2 = c 1 , c 3 = 2 c 1 and:

q = 1 (1 h ) ( 1 + 2 ) q T (54)
Using ( 18)-( 20), we obtain that:

h = [(1 h ) ] 1 1 ( 1 + 2 ) 1 h T w = (1 ) [ (1 h ) ( 1 + 2 )] 1 w T
and using ( 43)-(45), we get:

x 2 = 2 1 + + 2 2 2 + 1 1 (1 w ) w T x T 2 x 1 = B 1 (1 w ) w T x T 1 x T 1 + x T 2 = + 2 2 1 + + 2 1 1 + 2 2 1 + 2 (1 w ) w T
When = 1, the two tax rates are identical. In this case, they do not a¤ect the existence of the bubble, as follows from the expression of x T 1 + x T 2 . Thus, the existence of a bubble does not depend on the level of the two tax rates, but on the di¤erence between them. More precisely, a bubble (x T 1 + x T 2 > 0) exists if:

1 < 1 + 2 2 2 + 1 + 2 2 1 + + 2 (55)
Moreover, a bubbly steady state is unconstrained if

Rx T 1 > (1 h )q T h T .
Substituting x T 1 ; q T and h T in this expression, we deduce that the credit constrained is not binding at this steady state if and only if:

> 1 B 1 1 + 2 1 T (

56)

Proposition 9 There is a unique unconstrained bubbly steady state with taxation h T if:

1. > T ; 2. 1 < 1 + 2 2 2 + 1 +2 2 1+ + 2 .
Note that this proposition is a direct modi…cation of Proposition 2, aimed to include the e¤ect of taxes on the existence of an unconstrained bubbly steady state. As a main insight, we show that the e¤ect of taxes on the existence of this steady state does not depend on the level of the tax rates, but on the di¤erences between the two tax rates, measured by the parameter : More precisely, the bubble exists if is small, which requires that either h is large or w is small. When h is large, …scal policy introduces the incentives to invest in the bubble. When w is small, wages after taxes are large, which limits the amount of borrowing needed to invest in human capital. This facilitates the existence of an unconstrained bubbly equilibrium.

The e¤ect of …scal policy on the occurrence of a productive bubble

We …rst examine the coexistence of the constrained bubbleless and the unconstrained bubbly steady states with taxation.

Proposition 10

The unconstrained bubbly steady state with taxation (h T ) and the constrained bubbleless steady state with taxation (h T ) coexist if the following conditions are satis…ed:

1. 2 1 < ; 2. T < < e T ; 3. 1 + 2 1 1+ + 2 < 1 < (1+2 ) 1+ + 2 1 + 2 1 +2 2 .
Proof. See Appendix A.6.

We examine now whether the bubble is capital enhancing or productive. More precisely, q T < q T and h

T > h T if and only if:

F ( ) < 1 + 2 1 1 1 1 + 2 1 + + 2 (57)
where F ( ) is de…ned and has the same properties than in equation ( 40). The previous inequality and Proposition 10 imply that the existence of productive bubbles requires:

(1 + ) 1 + (1 + ) < 1 < (1 + 2 ) 1 + + 2 1 + 2 1 + 2 2
Using the same procedure than in Proposition 4, we show: Proof. See Appendix A.7.

11 If 2 1 <
We note the following direct implications of this proposition. First, the bubble may only be productive if it is characterized by x T 1 < 0; because 1 > (1+ ) 1+ (1+ ) . Thus, the bubble is productive only when it is used to …nance human capital investment. Second, using (50) and (54), we have:

q T q T = ( 1 + 2 ) 1 ( 1 + 2 ) (1+ ) 1+ + 2 + 1 (1 ) 1 2 + (1+ ) 1 (58)
We deduce that the gap between q T and q T decreases with and, hence, the gap between h T and h T increases with . Third, we can easily show that both

T and e T are increasing in , whereas b T is decreasing (because F 0 ( b T ) < 0). This means that following an increase of , the interval to have an unproductive bubble ( T ; b T ) shrinks, while the one to have a productive one ( b T ; e T ) enlarges. Therefore, given , a higher can allow a bubble to become productive.

The previous implications show that the productive role of the bubble is determined by the relative signi…cance of the tax rate on human capital h and on labor income w . The reason is quite immediate. As we have seen above, human capital at the bubbleless steady state is determined by the wage income and the amount of collateral, and therefore, decreases with both the labor and the capital tax rates. In contrast, the level of human capital at the bubbly steady state is determined by an arbitrage condition between the speculative asset and capital. This explains that human capital at the bubbly steady state is only a¤ected by the capital tax rate. We understand from these two observations that the bubble is capital enhancing if is large enough, i.e. the human capital tax rate is low enough and/or the labor tax rate is su¢ ciently large.

The …scal policy parameter determines the e¤ect of …scal policy on the existence of bubbles and its nature, productive or unproductive, and it also determines the existence of a constrained bubbleless steady state. On the one hand, we have shown that for intermediate values of , a productive bubble may arise because the after-tax return on human capital is not too large to rule out bubble holding in middle age and the after-tax wage is large enough to sustain a large enough investment in human capital when young and the purchase of the bubble in middle age. In this case, the bubble is used by the young to …nance human capital and by the middle age to transfer wealth to the future. On the other hand, recall that if is too large, there is no bubble in the economy.

From the previous analysis, we obtain two interesting insights for the …scal policy design. First, decreasing capital taxation is not always an appropriate policy to boost productive investment. If the policy is already accommodating, it may rule out the bubble and eliminate a mean to …nance human capital. Second, reducing the labor income tax may not be growth enhancing when there is already a bubble. In this case, this policy may change the nature of the bubble making it unproductive. We conclude that the suitable …scal policy to promote high levels of human capital and, therefore, of GDP depends on the existence and nature of the bubble.

Concluding remarks

This paper analyzes the interplay between liquid speculative bubbles with returns in the short term and less liquid productive investments in human capital which give returns in a longer term. We introduce this temporal distinction using an overlapping generations model with three-period lived agents. Agents make a portfolio choice between investment in human capital that gives returns during two periods, credit and a bubble traded at each period of time. Agents investments are limited by a credit constraint, where human capital plays the role of collateral. Considering that the bubbleless economy is dynamically ef-…cient and credit constrained, the bubble enhances production because young traders, who invest in human capital, are short sellers of the speculative asset and are no more constrained by the limit on credit, while middle age traders buy the bubble. As a direct implication, we analyze the e¤ect of …nancial regulation through the credit constraint on the levels of capital and bubble.

Our framework allows us to discuss the e¤ect of permanent technological shocks. Productivity shocks either biased toward the short term return or neutral with respect to the relative productivities of capital in the short and longer terms push up human capital investment, production and the bubble size. Given that these permanent shocks can be interpreted as innovations, this result explains the connection between periods of innovations and the raise of both production and bubbles, which is found in the empirical literature. However, we show that this connection does not happen if the technological shock mainly increases the productivity of capital in the long run.

Introducing …scal policy, we precisely discuss the e¤ects of taxation of human capital versus labor income on the level of human capital and the bubble size. We argue that a low …scal pressure on human capital income relative to labor income can be bad for productive investments, as this …scal policy may rule out the bubble.

Di¤erentiating this equation with respect to 1 , we obtain:

@h @ 1 1 h (1 + ) 1 + + 2 ( 1 + 2 ) 1 + 2 1 (1 ) 1 (1 + ) + 2 = (1 + ) 1 + + 2 1 + 2 1 (1 ) h 1 (1 + ) + 2 2 (1 ) 1 1 + 2 ( 1 (1 + ) + 2 ) i (1 )[ 1 (1 + ) + 2 ] 2 > 0
Now, let 2 = a 1 , with a a positive constant. Substituting this expression of 2 in (65), we easily get:

@h @ 1 1 h 2 =a 1 = 1 > 0 Finally, @h @ 2 2 h (1 + ) 1 + + 2 ( 1 + 2 ) 1 + 2 1 (1 ) 1 (1 + ) + 2 = (1 + ) 1 + + 2 2 2 1 2 (1 ) (1 )[ 1 (1 + ) + 2 ] 2 1 + 1 1 + 2 ( 1 (1 + ) + 2 )
which has an undetermined sign under the assumptions

A.5 Proof of Proposition 6

From (34) and (35) we deduce that @q=@ 1 < 0, @h=@ 1 > 0, @q=@ 2 < 0, @h=@ 2 > 0 and, taking 2 = 1 constant, @q=@( 1 + 2 ) < 0 and @h=@( 1 + 2 ) > 0.

We now focus on the bubble size and speculative asset holdings. The wage is given by w = ( 1) 1 ( 1 + 2 ) 1 . Therefore, using (37) and inequality (41), we easily deduce that @x 1 =@ 1 < 0; @x 1 =@ 2 < 0 and, taking 2 = 1 constant, @x 1 =@( 1 + 2 ) < 0.

Note that 2 =( 2 + 1 ) is decreasing in 1 and does not vary when 2 = 1 stays constant. Note also that inequality (41) implies that x 1 < 0 and, hence, x 2 > 0: Then, by direct inspection of (36), we get @x 2 =@ 1 > 0 and, taking 2 = 1 constant, @x 2 =@( 1 + 2 ) > 0. Substituting the expression of the wage in (36), we also get:

@x 2 @ 2 = 1 1 ( 1 + 2 ) 2 1 1 2 1 + 2 1 2 1 1 + 1 + + 2
This is strictly negative if and only if

(1 )(1 + ) > (1 2 ) 2 (1 + ) 2 1 29
This last inequality is satis…ed for all 2 = 1 > 0 if its right-hand side is negative. It requires:

1 > 2 1 + + 2
Since this is always satis…ed under inequality (41), we have @x 2 =@ 2 < 0. Using all these results, we obviously have @(x 1 + x 2 )=@ 2 < 0. Let us note that (2 2 + 1 )=( 2 + 1 ) decreases with respect to 1 and does not vary when

B Appendix B B.1 Constrained bubbly steady state

The market clearing condition [START_REF] Hirano | Asset bubbles, endogenous growth, and …nancial frictions[END_REF] implies that R c = 1. From the …rst order conditions ( 12)-( 14) and the budget constraints, we get:

h = 1 1 + 2 q c 1 1 h c (68) q = 1 1 ( 1 + 2 ) (1+ ) 1+ (1+ ) + 1 q c (69)
and

x 1 = 1 q c h c x c 1 (70) x 2 = (1 ) 1 2 1 + h c q c x c 2 (71) x c 2 + x c 1 = [ (1 + 2 )] 1 2 1 + q c h c (72)
We deduce that there is a positive bubble x c 2 + x c 1 > 0 if and only if < and 2 = 1 < , where is given by [START_REF] Wehinger | Fostering long-term investment and economic growth, summary of a high-level OECD …nancial roundtable[END_REF].

The credit constraint is binding if c 2;t+1 > R t+1 c 1;t . At a bubbly steady state, this condition holds if

< 1 + 2 1 1 1 (1 + ) 1 + (1 + ) that requires 1 > (1+ ) 1+ (1+ ) .
Proposition 12 There is a unique constrained bubbly steady state h if:

1. < minf ; g; 2. 2 = 1 < ; 3. 1 > (1+ ) 1+ (1+ ) .
We next show that the bubble cannot be productive when both bubbleless and bubbly steady states are constrained. We claim that a bubble is productive if h c > h or, equivalently, q c < q. Using ( 28) and (69), we get that the bubble is productive if > : This contradicts the existence of a positive bubble. Using Propositions 1 and 12, we deduce that:

Proposition 13 The constrained bubbly (h c ) and bubbleless (h) steady states coexist if:

1. < minf ; e ; g;

2. 2 = 1 < ;

3. 1 > maxf (1+ ) 1+ + 2 ;1 + 2 1 1+ + 2 g.

In addition, we have h c < h.

B.2 Positive probability of bubble crash

We aim to understand whether our results still hold when there is a positive probability of market crash for the bubble. To this end, we extend our framework to the case where the bubble is stochastic. When there is a bubble, agents may coordinate their expectations on an equilibrium without bubble because of a sunspot process which associates a positive probability to a market crash. In such an economy, a stochastic bubbly steady state, i.e. a steady state with positive bubble that takes into account that the bubble may crash with a positive probability, will coexist with the bubbleless one examined in Section 3. Following [START_REF] Weil | Con…dence and the real value of money in overlapping generations models[END_REF], we consider a Markov process of a bubble crash. If there is no bubble at period t, there is no bubble at period t + 1 with a probability 1. If there is a bubble at period t, there is a probability 2 (0; 1] such that the bubble persists at the next period and a probability 1 such that the bubble crashes at period t + 1. Note that a market crash in period t + 1 means that the price p t+1 of the asset without fundamental value is zero, i.e. b 1;t+1 = b 2;t+1 = R t+1 = 0 using the notations of Section 2. In contrast, credit d t+1 and its return R d t+1 remain positive, even if they are a¤ected by the bubble crash.

Let us examine the household's behavior in such a stochastic environment. To …x ideas, we focus on a household born at period t and we assume that there is a bubble at this date. 18 In the following, we denote c + 2;t+1 (c 0 2;t+1 ) the consumption in middle age when the bubble persists (crashes) in t + 1. If the bubble crashes in t + 1, the consumption when old is c 00 3;t+2 . If the bubble does not crash in t + 1, the consumption when old is c ++ 3;t+2 if it does not crash in t + 2 too, while it is c +0 3;t+2 if it crashes in t + 2 (see also Figure 3). Using ( 2)-( 4), we deduce that the di¤erent household's consumptions are given by: c 1;t = w t + d t h t+1 b 1;t ;

(73) c + 2;t+1 where variables without the zero as a superscript correspond to choice variables decided when there is a bubble and resulting prices. In contrast, d 0 t+1 is the choice of debt holding following a bubble crash. Moreover, even if human capital is a long term investment, q 0 t+2 in (78) is di¤erent from q t+2 because it depends on h 0 t+2 , which is the investment in human capital decided by a new generation after the bubble crash. Finally, R d;0 t+2 is the return of debt in the absence of bubble.

= 1 q t+1 h t+1 + R
The household determines her choices maximizing her expected utility E t [u (c 1;t ) + u (c 2;t+1 ) + 2 u (c 3;t+2 )], that rewrites: u(c 1;t ) + u(c + 2;t+1 ) + ( 1)u(c 0 2;t+1 ) + where R d;0 t+2 = R t+2 . To be consistent with the analysis of Section 3, the credit constraint should be binding when the bubble crashes, i.e. d 0 t+1 = 1 q t+1 k t+2 =R t+2 . Moreover, when the bubble is positively valued at period t, the credit constraint should not be binding, i.e. R d t+1 d t < 1 q t+1 k t+1 + R t+1 b 1;t .

The …rst order conditions (80)-( 85) and the budget constraints (73)-( 78) can be used with the prices ( 19)-( 20) and the equilibrium conditions on the asset markets to determine a stochastic bubbly equilibrium. When is equal to 1, equations ( 80)-( 85) are equivalent to the …rst order conditions without bubble crash (59)-(61) (with t = 0), which means that the allocation corresponds to the deterministic bubbly equilibrium. By continuity with respect to , the equilibrium will still exist when is smaller but su¢ ciently close to 1. This is especially true when we focus on the stationary allocations, as in Section 3. This is similar to what we can …nd in the seminal paper by [START_REF] Weil | Con…dence and the real value of money in overlapping generations models[END_REF], and many more recent papers like Fahri and Tirole (2012), [START_REF] Hirano | Asset bubbles, endogenous growth, and …nancial frictions[END_REF] or [START_REF] Kocherlakota | Bursting bubbles: consequences and cures[END_REF]. Since the introduction of a positive probability of bubble burst will not bring any new ingredient for our explanation of productive bubble at the cost of strong additional complexities, we do not explicitly solve the stochastic model for the sake of conciseness.

Proposition 5 that 2 1 <

 51 Assuming , b < < e and 1 strictly lower but su¢ ciently close to(1+2 )

Figure 1 :

 1 Figure 1: E¤ect of permanent technological shocks

Figure 2 .

 2 Figure 2. Transitional dynamics after a permanent technological shock.

Figure 3 :

 3 Figure 3: Consumption pro…le when the bubble has a probability to crash 1

  under the constraints (73)-(78) with respect to b 1;t , b 2;t+1 , d t , d t+1 , d 0 t+1 and h t+1 , we obtain:

"

  and 1 is strictly lower but su¢ ciently close to

	(1+2 ) 1+ + 2	1 + 2 1 +2 2	, we have:
	1. The unconstrained bubbly steady state with taxation has a lower level of
	capital than the constrained bubbleless steady state with taxation (h	T <
	h T ), i.e. the bubble is unproductive, if 2 ( T ; b T	);
	2. The unconstrained bubbly steady state with taxation has a higher level of
	capital than the constrained bubbleless steady state with taxation (h	T >
	h T ), i.e. the bubble is productive, if 2 ( b T	; e T	).

  t+1 b 1;t R d t+1 d t b 2;t+1 d t+1 ;

			c 1;t	H H H H H	H H H H H 2;t+1 c +
					(74)
	c 0 2;t+1	=	1 q t+1 h t+1 R d t+1 d t d 0 t+1 ;	(75)
	c ++ 3;t+2	=	2 q t+2 h t+1 + R t+2 b 2;t+1 + R d t+2 d t+1 ;	(76)
	c +0 3;t+2	=	2 q t+2 h t+1 + R d t+2 d t+1 ;	(77)
	c 00 3;t+2	=	2 q 0 t+2 h t+1 + R d;0 t+2 d 0 t+1	(78)

There are however some exceptions. See for instance[START_REF] Kamihigashi | The spirit of capitalism, stock market bubbles and output ‡uctuations[END_REF].

Other examples of productive investments with a long term return are real estate, infrastructure, equipment and software, or research and development.

[START_REF] Bosi | On rational exuberance[END_REF] In the middle age, agents do not invest in the productive asset because, at the equilibrium with bubble we focus on, they will prefer to reallocate income across generations using the speculative bubble.

Note that in models with in…nitely lived agents, short sale positions cannot sustain the existence of a bubble. See[START_REF] Kocherlakota | Bubbles and constraints on debt accumulation[END_REF] for more details.

Note that we can …nd an alternative approach of liquidity in[START_REF] Lagos | A uni…ed framework for monetary theory and policy analysis[END_REF] and Lagos (2010).

Taxes on human capital income must be interpreted as the tax rate on high wages, which are typically associated with high skills (human capital). In contrast, taxes on labor income must be interpreted as the tax rates on low (unskilled) wages.

As it is well-known, this idea was already emphasized in the model without capital accumulation by[START_REF] Samuelson | An exact consumption-loan model of interest with or without the social contrivance of money[END_REF].

[START_REF] Diamond | Bank runs, deposit insurance and liquidity[END_REF] See[START_REF] Miao | Introduction to economic theory of bubbles[END_REF] for a short survey of this recent literature.

Short sales on the speculative asset imply b 1;t < 0: In this case, young individuals use the speculative asset to …nance investment.

0 See Appendix A.1 for more details. Note also that we focus on deterministic equilibria, since we do not directly consider the crash of the bubble as a stochastic process. As explained in Appendix B.2, the introduction of a stochastic process does not provide new additional ingredients to explain the occurrence of productive bubbles.

[START_REF] Benhabib | Vintage capital, investment and growth[END_REF] We limit the analysis to the steady state equilibrium. However, we have numerically obtained the transition, which exhibits monotonic convergence towards the steady state. Section 4 shows this transition.

[START_REF] Bosi | On rational exuberance[END_REF] In order to obtain the condition for dynamic ine¢ ciency, we …rst use the resource constraint to obtain that total consumption, ct; satis…es ct = [ 1 ht

+ 2 h t 1 ] h t+1 : Using this equation, the introduction of a permanent reduction of investment in period t, dh t+1 = dh t+j < 0 for all j 1; has the following e¤ects on total consumption: dct = dh t+1 ; dc t+1 = ( 1 q t+1 1) dh t+1 and dc t+j = [( 1 + 2 ) q t+j 1] dh t+1 for all j 1: From these expressions, it follows that if ( 1 + 2 )q < 1 then a permanent reduction in investment implies an increase in total consumption in all dates and, hence, the equilibrium is dynamically ine¢ cient.

; in the same proportion;

x 2 and x 1 + x 2 increase if there is an increase of either 1 or of both, 1 and 2 ; in the same proportion, but they decrease if there is an increase of 2 .

[START_REF] Boucekkine | Obsolescence and modernization in the growth process[END_REF] Note that Panel (a) is drawn considering that h is increasing in

.1 6 Following[START_REF] Shiller | Irrational Exuberance[END_REF],[START_REF] Lansing | Speculative bubbles and overreaction to technological innovation[END_REF] relate four major run-ups in the real S&P 500 stock index with the following technological advances: (i) In 1900s, the high speed rail travel, transatlantic radio and long-line electric transmission; (ii) In 1920s, mass production, travel

= 1:5 and

= 0:0875 and after the shock 1 = 1 and 2 = 0:35: These changes in the parameters are aimed to increase the ratio 2 = 1 ; which makes the bubble disappear, while

= 1 stays constant. Using (38), we deduce that @(x 1 + x 2 )=@ 1 > 0 and, taking 2 = 1 constant, @(x 1 + x 2 )=@( 1 + 2 ) > 0.

[START_REF] Diamond | Bank runs, deposit insurance and liquidity[END_REF] If there is no bubble at this period, one get the bubbleless economy.

This work has been carried out thanks to the support of the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, the …nancial support of the French National Research Agency (ANR-15-CE33-0001-01) and the ECO2015-66701-R (MINECO-FEDER, UE) grant funded by the Government of Spain and Fondo Europeo de Desarrollo Regional. We thank an Editor and two Referees for their comments that really improve the paper. We also thank Raouf Boucekkine, Lise Clain-Chamosset-Yvrard, Michel Lubrano and participants to the 5th AMSE

A Appendix A A.1 Household' s program

Households maximize the utility (1) subject to ( 6)- [START_REF] Diamond | Bank runs, deposit insurance and liquidity[END_REF] and the constraint [START_REF] Fahri | Bubbly liquidity[END_REF]. Let t be the Lagrangian multiplier associated to equation [START_REF] Fahri | Bubbly liquidity[END_REF]. Then, the …rst order conditions with respect to x 1;t ; x 2;t+1 and h t+1 are, respectively, u 0 (c 1;t ) + u 0 (c 2;t+1 ) R t+1 + t R t+1 = 0 (59) u 0 (c 2;t+1 ) + 2 u 0 (c 3;t+2 ) R t+2 = 0 (60) u 0 (c 1;t ) + u 0 (c 2;t+1 ) 1 q t+1 + 2 u 0 (c 3;t+2 ) 2 q t+2 + t 1 q t+1 = 0 (61)

From (59), we obtain that t = u 0 (c 1;t ) =R t+1 u 0 (c 2;t+1 ) : Therefore, if u 0 (c 1;t ) =R t+1 > u 0 (c 2;t+1 ), then t > 0 and the constraint is binding. Taking into account that u(c j;t ) = ln c j;t , the solution of the household's problem when the credit constraint is binding is given by equations ( 12)- [START_REF] Kamihigashi | The spirit of capitalism, stock market bubbles and output ‡uctuations[END_REF].

If the credit constraint is not binding, we have t = 0 and the solution of the household's problem is given by equations ( 15)- [START_REF] Lagos | Asset prices and liquidity in an exchange economy[END_REF].

A.2 Proof of Proposition 3

Using Proposition 1, we recall that there exists a constrained bubbleless steady state if < minf ; e g, 2 = 1 < and 1 > 1 + 2 1 1+ + 2 . Using Proposition 2, there exists a unique unconstrained bubbly steady state if > and 1 <

. Using ( 30) and (39), we obtain that > if and only if

. Substituting in [START_REF] Weil | Con…dence and the real value of money in overlapping generations models[END_REF], we see that it holds as a strict inequality if and only if 1

. We deduce that, in this case, we have < e . Substituting [START_REF] Wehinger | Fostering long-term investment and economic growth, summary of a high-level OECD …nancial roundtable[END_REF] in [START_REF] Weil | Con…dence and the real value of money in overlapping generations models[END_REF], we see that this last inequality never holds under 1

, which means that > e . Finally, we

is satis…ed under 2 = 1 < . The proposition immediately follows.

A.3 Proof of Proposition 4

In the main text, we show that h > h if condition (40) holds. We can see that F ( ) is an inverse U-shaped concave function (F ( ) 00 < 0) with F (0) = F (1) = 0, F 0 (0) > 0 and F 0 (1) < 0. Substituting in inequality (40), we can show that the inequality is not satis…ed when 1

. Using (31), we get:

e (1 e )

In addition,

is equivalent to e < , which is satis…ed, as shown in the proof of Proposition 3. This means that for 1 lower but su¢ ciently close to (1+2 )

, e satis…es inequality (40). Therefore, there exists a unique b 2 ( ; e ) such that:

We deduce that inequality (40) is satis…ed for all 2 ( b ; e ) and it is not satis…ed for 2 ( ; b ), which proves the results of Proposition 4.

A.4 Proof of Proposition 5

From ( 28) and ( 29), we obtain

A.6 Proof of Proposition 10

First, note that we have

1+ + 2 . We can then prove the results in Proposition 10 from the analysis in Section 3.

A.7 Proof of Proposition 11

We can show that for = T , inequality (57) is not satis…ed under 1 <

. Moreover, using (51), we have:

We deduce that F ( e T ) satis…es inequality (57) if and only if:

Then, there exists b T 2 ( T ; e T ) such that inequality (57) is not satis…ed for 2 ( T ; b T ), but is satis…ed for 2 ( b T ; e T ). We also note that the right-hand side of this inequality is strictly lower than