Lise Clain-Chamosset-Yvrard

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

Aurélien Eyquem

Frederic Jouneau-Sion

Bubble on real estate: The role of altruism and fiscal policy

Keywords: JEL classification: E22, E44, G11 Bubble, Altruism, Real estate, Credit, Overlapping generations

In this paper, we are interested in the interplay between real estate bubble, aggregate capital accumulation and taxation in an overlapping generations economy with altruistic households. We consider a three-period overlapping generations model with three key elements: altruism, portfolio choice, and financial market imperfections. Households realise different investment decisions in terms of asset at different periods of life, face a binding borrowing constraint and leave bequests to their children. We show that altruism plays a key role on the existence of a productive real estate bubble, i.e. a bubble in real estate raising physical capital stock and aggregate output. The key mechanism relies on the fact that a real estate bubble raises income of retired households. Because of higher bequests, there children are able to invest more in productive capital. Introducing fiscal policy, we show that raising real estate taxation dampens capital accumulation.

Introduction

The question we address in this paper is how we can explain that a real estate bubble can enhance productive capital and therefore aggregate output, whereas we could have in mind that investments in productive capital and in real estate could rather be substitutable. This question is important regarding the consequences of the last financial crisis and also for the design of the fiscal policy. On the one hand, many countries have experienced real estate bubbles before the last financial crises and, at the same time, a rise in capital stocks and aggregate output. Some empirical studies highlight that private non-residential investment in the United States increased significantly during the formation of the real estate bubble in the 2000s and then dropped when it bursts (IMF, 2015). 1On the other hand, we note that the fiscal policy is especially designed to guide household savings. The french 2018 Finance Act (Loi de finances 2018), which replaces the solidarity tax on wealth by a tax on real estate wealth, is a perfect illustration. By concentrating taxes on real estate, considered as unproductive, a government aims to divert savings allocated from unproductive investment to productive one in order to promote economic growth. As wealth mainly consists in real estate for a large part of households, and most of them invest in real estate to leave an inheritance to their children, this idea that taxing real estate will necessarily promote productive capital could not hold for the whole economy.

The aim of this paper is to exhibit a mechanism through which a rational real estate bubble is accompanied by a boom in physical capital stock and in aggregate output. Our explanation is based on the altruistic behaviour of households. Our framework will be further used to study the macroeconomic consequences of real estate taxation.

We consider an overlapping generations model with three-period lived households. Households realise different investment decisions in terms of asset at different periods of life. When young, households can borrow and invest in real estate by buying land or a house for their old age. When middle-aged, they can save in the forms of deposits and physical capital to transfer wealth to their old age.

In our paper, there are three key elements. First, assets play different roles in our model. In addition to serve as a collateral, real estate allows young households to transfer resources to their old age. Accordingly, real estate can be seen as a long-term investment. Households behave as investors who buy a real estate asset for resale purpose only, and not for housing services. The real estate asset is intrinsically useless, meaning that its fundamental value is zero. This asset is a bubble when it has a positive value. Deposits and physical capital are both short-term investments. Indeed, households invest in these assets when middle-aged and receive returns when old. Nevertheless, deposits are the counterpart of loans and capital is used in production. This segmentation of asset markets reflects the idea that households first invest in real estate, kept during all the life-cycle and used to transfer bequests to children. Once this investment is reimbursed, agents are concerned with financial investments, through deposits and productive capital.

Second, we take into account financial market imperfections through a binding borrowing constraint faced by young households. This contrasts with most of the recent contributions in this literature, which focus on financial frictions at the entrepreneur level (among others, Farhi and Tirole, 2012; Martin and Ventura, 2012; [START_REF] Miao | Housing bubbles and policy analysis[END_REF][START_REF] Hirano | Asset bubbles, endogenous growth, and financial frictions[END_REF]. In addition, some empirical studies support our assumption (see [START_REF] Campbell | Consumption, income and interest rates: Reinterpreting the time series evidence[END_REF][START_REF] Jappelli | Who is credit constrained in the US economy?[END_REF][START_REF] Crook | Household debt and credit constraints: Evidence from OECD countries[END_REF]. In our paper, households must pledge their labour income and their house as collaterals and, borrow against a part of their labour income and a part of the market value of their real estate asset. 2 An increase in the market value of the real estate asset or a higher labour income help young households to relax their borrowing constraint. In this paper, labour income can be seen as a fundamental collateral and the real estate asset as a bubbly one. 3The first type of collateral would rather represent the french system, while the second one would rather describe what happens in US. In addition, it will allow us to discuss the respective role of each type of collateral on the existence of a real estate bubble associated to a higher level of capital and output.

Third, households behave altruistically by leaving bequests to their children. Several empirical studies support the idea that households have a desire to leave bequests and, underline its importance for explaining saving behaviour [START_REF] Kotlikoff | The role of intergenerational transfers in aggregate capital accumulation[END_REF][START_REF] Dynan | Do the rich save more?[END_REF]). In our model, the desire to leave bequests are rationalized by a family altruism, defined as the situation in which parents care about their children's income. As we will see, altruism is crucial for our results.

Our main concern is to investigate and understand the existence of a productive real estate bubble, i.e. a real estate bubble promoting capital accumulation and, thus aggregate output. We show that a real estate bubble occurs in a low interest rate environment. In this case, there exists an excess of savings in the economy, which provides a room for alternative stores of value. As the real estate bubble provides a return equal to one at the steady state, this return creates an incentive for households to hold the bubble and borrow to invest in this bubble.

Comparing the bubbleless and bubbly steady states, we show that a productive real estate bubble occurs when parents are sufficiently altruistic and the effects of fundamental and bubbly collaterals coming from the borrowing constraint are not too high. This result is consistent with empirical evidence showing that productive investment increases significantly during the formation of a bubble [START_REF] Martin | Economic growth with bubbles[END_REF]. Altruism plays a key role on the occurrence of productive real estate bubbles. In an economy without altruism, middle-aged households have not a high enough income to sustain a higher level of capital in presence of a real estate bubble. Moreover, a productive real estate bubble exists whatever the type of collateral (fundamental or bubbly collateral) required by lender on the credit market. Surprisingly, the two types of collateral play quite similar roles and we cannot conclude that one type further promotes the emergence of real estate bubbles.

Finally, we also investigate the effect of fiscal policy on the stock of physical capital and, hence, on aggregate output. We show that an increase of a real estate tax has a negative effect on productive investment. An increase of the tax rate reduces the incentive to buy the real estate bubble, which reduces the income when old. It implies a negative effect on bequest distributed to middle-aged households, who invest less in physical capital. This result questions the view arguing that a government should increase taxation of unproductive assets, like real estate taxation, in order to boost productive investment and, thus aggregate output.

The paper proceeds as follows. In the next section, we discuss the related literature. Section 3 presents the model. Section 4 characterizes the co-existence of steady states without and with a real estate bubble and shows that the real estate bubble may enhance capital. Section 5 is devoted to the presentation of the model without altruism. Section 5 discusses the role of fiscal policy. A last section provides concluding remarks, whereas technical details are relegated to an Appendix.

Related literature

Our paper is related to a recent growing literature on rational housing bubbles in overlapping generations (OLG, henceforth) models. The existence of such bubbles has been studied in OLG models first by Arce and Lopez-Salido (2011) and [START_REF] Basco | Globalization and financial development: A model of the dot-com and the housing bubbles[END_REF]. They investigate the interplay between real estate bubbles and financial frictions, but without physical capital accumulation in contrast to our paper. In their models, households deriving utility from housing services, but they model the housing bubble as a shortage of assets in the economy, and not as a component of the housing price as in our model. By the way, Arce and Lopez-Salido (2011) show that the housing price is less than its fundamental value, i.e. the sum of discounted dividends, in the bubbly equilibrium. [START_REF] Zhao | Rational housing bubble[END_REF] builds an OLG model with homeowners, who derive utility from housing services, and investors, who do not. He shows that a housing bubble only exists in an equilibrium in which investors hold houses for resale purpose only, and not for dividend purpose either in terms of rents or in terms of utility. In a recent contribution, [START_REF] Huber | Preference for housing services and rational house price bubbles[END_REF] shows that an housing bubble, as a component of the housing price, can exist in an OLG economy with housing services, but without capital accumulation. Her results does not rely on the existence of any financial frictions, but on the features of the housing market. At each period, the young generation is endowed with a part of the housing stock belonging to the old generation. Through this feature, she introduces a new bubble at each period. In contrast to [START_REF] Zhao | Rational housing bubble[END_REF] and Huber (2018), we abstract ourselves from modelling housing services through utility. This allows us to deeply focus on the the role of altruism in the existence of rational real estate bubble and on the interplay between rational real estate bubble and capital accumulation.

To the best of our knowledge, only [START_REF] Bosi | Financial bubbles and capital accumulation in altruistic economies[END_REF] investigate the influence of altruism on the existence of a bubble. They consider a two-period OLG model with altruism and an asset bringing non-stationary positive dividends. They study the dynamics of capital stock and asset value. In contrast to our paper, they consider a form of altruism rationalized by a bequest constraint and are not concerned with the design of fiscal policy.

This paper also belongs to the rational bubble literature studying the macroeconomic effects of a bubble in OLG models with financial frictions. Farhi and Tirole (2012) and [START_REF] Martin | Economic growth with bubbles[END_REF] show that a bubble can have a crowd-in effect by promoting capital accumulation, and thus increasing aggregate output through the existence of financial frictions, which generates a wealth effect coming from the bubble. Even if we consider financial frictions, we differ from Martin and Ventura (2012) because our analysis does not require bubble shocks, and from Farhi and Tirole (2012) because our mechanism is not based on the existence of a wealth effect. It relies on the existence of altruistic households and, thus on inheritance. By receiving bequests, middle-aged households have a sufficiently high income to sustain a higher level of capital during bubbly episodes. [START_REF] Raurich | On the interplay between speculative bubbles and productive investment[END_REF] investigate the interplay between long-term productive investments in physical capital and short-term speculative ones in the asset bubble. Their mechanism is also different than ours, since it relies on selling short the bubble to invest more in capital. There is also a large literature studying the existence of productive bubbles in infinite horizon models which differs from our approach. For instance, [START_REF] Kocherlakota | Bursting bubbles: Consequences and cures[END_REF], Hirano and Yanagawa (2017) and Miao and Wang (2018) build models with credit market imperfections. In contrast, [START_REF] Kamihigashi | The spirit of capitalism, stock market bubbles and output fluctuations[END_REF] shows that bubbles can affect output positively in an economy with increasing social returns and capital spirit.

Finally, only few other contributions analyse the effects of taxation on both the existence of bubbles and on capital accumulation. [START_REF] Miao | Housing bubbles and policy analysis[END_REF] are interested in the existence of a real estate bubble in an infinite-horizon model with production and credit constraints faced by the entrepreneurs. They show that a bubble can exist and reduce welfare. Some fiscal policies, such as property or transaction taxation, are able to eliminate the bubble. We differ from this paper since we consider a model with altruistic households facing credit constraints. Furthermore, we address a different political issue. Raurich andSeegmuller (2017, 2019) also deal with fiscal policy issue, by analyzing the consequences of capital and labour income taxes on the existence and macroeconomic effect of a bubble. In our paper, we consider an alternative source of taxation, namely a real estate taxation.

Model

We consider an overlapping generations model with discrete time (t = 0, 1, ..., +∞). This economy consists of two types of agents, households and firms.

Households

There is no population growth and, at each date t, a generation of unit size is born. Each generation lives for three periods.

Utility. Preferences are represented by an additively separable life-cycle utility function:

lnc 1,t + βlnc 2,t+1 + β 2 lnc 3,t+2 + δ ln(φw t+2 + x t+2 ) (1)
where c 1,t , c 2t+1 , and c 3,t+2 denote the consumption at time t, t + 1, t + 2 of a household born at t, β ∈ (0, 1) the discount factor and δ > 0 the degree of altruism of a household. The parameter ∈ {0, 1} allows us to consider the model with or without altruism. Through these preferences, we introduce family altruism, defined as the situation in which generation t cares about the next generation income. More precisely, old parents take care about the income of their middle-aged children which has two components: x t+2 is her bequest received from their old parents born at time t and φw t+2 the labour income of a middle-aged agent born at time t + 1, where φ > 1 represents the efficient units of labour inelastically supplied at the second period of life and remunerated at the wage w t+1 .

Budget constraints. In her first period of life, the household is young and supplies one unit of labour inelastically remunerated at the wage w t . She can borrow an amount d y t on a credit market. With her wage and borrowing, she can consume an amount c 1,t of final good and buy an asset h t+1 , which is assumed to be intrinsically useless, at the price p t in units of consumption. We interpret this asset as real estate. A positive price p t > 0 depicts a bubble on real estate.

At the second period of life, the household is middle-aged. As we have previously explained, she supplies inelastically φ efficient units of labour remunerated at the wage w t+1 . A middle-aged household may receive a bequest x t+1 from their old parents. Bequests and labour income are used to repay their debt at the gross rate R d t+1 , to consume c 2,t+1 of final good, and to save through a diversified portfolio of physical capital k t+2 and deposits d m t+1 . At the third period of life, the household is old and retired. She uses the sales of their real estate asset bought at time t at price p t+2 and her remunerated savings from physical capital at the gross rate R t+2 and from her deposit at the gross rate R d t+2 to consume c 3,t+2 units of final good and give a positive bequest x t+2 to her children.

4 c 1,t + p t h t+1 = w t + d y t (2) c 2,t+1 + R d t+1 d y t + d m t+1 + k t+2 = φw t+1 + x t+1 (3)
c 3,t+2 + x t+2 = p t+2 h t+1 + R t+2 k t+2 + R d t+2 d m t+1 (4)
x t+2 0 (5)
Through these budget constraints, we assume that a household chooses first to invest in a long term asset, real estate, that she resells when hold, and has access to the productive capital only at middle-age. Interpreting k t+2 as shares of firms, it symbolizes the idea that households first invest in real estate, kept during all the life-cycle and used to transfer bequests to children. The loans to finance this investment are reimbursed in middle-age. Then, agents are concerned with financial investments, through deposits and productive capital.

As regards the structure of our model, the three-period model is close to a natural broad subdivision of the life-cycle, and is especially used in OLG models dealing with housing bubble (see [START_REF] Arce | Housing bubbles[END_REF][START_REF] Arce | Housing bubbles[END_REF][START_REF] Basco | Globalization and financial development: A model of the dot-com and the housing bubbles[END_REF]. As underlined by Arce and Lopez-Salido (2011), this structure tends to replicate the fact observed in the data that households borrow to buy a real estate asset in the early part of their working life, then start to accumulate financial assets after acquiring their real estate asset. Indeed, some empirical studies show that the stock market participation rates are very low among young households, increase during their working life to reach a peak around age 50 and may decrease during retirement (see among other Poterba and Samwick, 2001, and Ameriks and Zeldes, 2004). [START_REF] Poterba | Household portfolio allocation over the life cycle[END_REF] also show that investments in real estate display a hump-shaped pattern over the life-cycle. Furthermore, [START_REF] Cocco | Portfolio choice in the presence of housing[END_REF] highlights the key role of real estate asset in portfolio choice over the life-cycle. Since investments in real estate is often financed through a mortgage contract as underlined by [START_REF] Heaton | Portfolio choice and asset prices: The importance of entrepreneurial risk[END_REF], young households have a limited financial wealth to participate in stock market.

Borrowing constraint. Doing their investment in real estate, young households face a borrowing constraint:

R d t+1 d y t ≤ θ 1 p t+1 h t+1 + θ 2 φw t+1 (6)
with 0 ≤ θ 1 < 1 and 0 ≤ θ 2 < 1

Borrowing can only take place up to the point where the principal plus interest, i.e. R d t+1 d y t , is secured by a fraction θ 1 of the market value of real estate owned by the household at t + 1, and a fraction θ 2 of her wage received at the period of debt repayment. If households repudiate their loan obligations, the lenders can seize a fraction θ 1 of their real estate asset. This collateral constraint is consistent with standard lending criteria used in the mortgage market. This type of constraint generates two wealth effects one which is direct coming from future wage, and the other one which is indirect by modifying the relative price of the asset used as collateral. Through this constraint, we make a distinction between a "fundamental" collateral (future wages) and a "bubbly" collateral (value of bubble asset).

Optimal behaviour. Let us assume = 1, i.e. agents are altruistic. A household derives her optimal consumption choice (c 1,t , c 2,t+1 , c 3,t+2 ), her optimal portfolio choice (k t+2 , d y t , d m t+1 , h t+1 ), and her optimal bequest (x t+2 ) by maximizing her utility function (1) under her budget, bequest and borrowing constraints (2)- [START_REF] Cocco | Portfolio choice in the presence of housing[END_REF]. We restrict our attention to an economy in which the borrowing constraint is binding. The optimal behaviour of a household born at time t is summarized by the following equations (see also Appendix 8.1):

β 2 c 3,t+2 (p t+2 -θ 1 p t+1 R t+2 ) = 1 c 1,t p t -θ 1 p t+1 R t+1 (7) 1 c 2,t+1 = R t+2 β c 3,t+2 (8)
R d t+2 = R t+2 (9)
β 2 c 3,t+2 = δ φw t+2 + x t+2 (10)
θ 1 p t+1 h t+1 + θ 2 φw t+1 = R t+1 d y t ( 11
)
β 2 R d t+1 R t+2 c 1,t < c 3,t+2 (12)
x t+2 > 0 (13)

Because of the borrowing constraint, Eq. ( 7) depicts a modified intertemporal trade-off between the first and third period consumptions. It requires that the following inequality must be satisfied at each period t:

θ 1 p t+1 p t < R t+1 < 1 θ 1 p t+1 p t (14)
Since households face no borrowing constraints at their second and third period of life, Eq. ( 8) depicts the standard intertemporal trade-off between the second and third period consumptions. In their second period of life, households save through a diversified portfolio of deposits and physical capital. As the two assets are perfectly substitutes, they should provide the same return (Eq. ( 9)). Eq. ( 10) depicts the trade-off between the strictly positive bequests and third-period consumption. Since we restrict our attention to an economy with a binding borrowing constraint (Eq. ( 11)), inequality (12) should be satisfied.

Firms

The technology is characterized by the following aggregate production function:

Y t = K α t L 1-α t , with α ∈ (0, 1/2) ( 15
)
where L t is the total amount of efficient units of labour and K t is the stock of physical capital in the economy. Using a = K/L, the intensive production function is Y t /L t = a α t , and competitive factor prices are given by :

w t = (1 -α)a α t ≡ w(a t ) (16)
R t = αa α-1 t ≡ R(a t ) (17)
Let s ≡ (1 -α)/α(> 1). Using Eqs. ( 16) and ( 17), we obtain the ratio of factor prices:

w t R t = sa t (18)

Asset markets

The market of real estate asset is segmented in the sense that only young and old households can trade this asset h t . As already explained, young households buy the asset in order to sell it when they will be old. Households behave as investors who buy real estate for resale purpose only, and not for housing services. We abstract from housing services through preferences for two reasons. First, a real estate bubble can exist in an OLG model with households deriving utility from housing services (homeowners) and others doing not (investors). In such a framework, [START_REF] Zhao | Rational housing bubble[END_REF] shows that a real estate bubble exists in an equilibrium if investors hold houses for resale purpose only and not for dividend purpose either in terms of rents or in terms of utility. As we are interested in the interplay between bubble and capital accumulation, we restrict our attention to the behaviour of an investor and do not model the behaviour of an homeowner. 5 Second, we consider a utility function including altruistic preferences. Not modelling real estate asset in the utility function allows us to deeply focus on the role of altruism in the existence of a productive bubble. The supply of this asset is fixed and normalized to 2. Since this asset is intrinsically useless, and thus has no fundamental value, a bubble exists if p t > 0.

Deposits from middle-aged are used for loans of the young. This explains that loans and deposits provide the same rate R d t+1 .

Intertemporal equilibrium

At date t = 0, the initial old household and initial middle-aged household hold one unit of real estate. Therefore, at each period, only one unit of financial asset is exchanged, implying that h t+1 = 1. Loans being equal to deposits, at each period of time, we have

d y t = d m t ≡ d t .
Taking into account that the population size of a generation is normalized to one, labor measured in efficient units is equal to L t = 1 + φ. Then, equilibrium on the physical capital market means that capital per capita is given by k t = (1+φ)a t . In addition, since physical capital and deposits are perfect substitutes for a household, they provide the same return at the equilibrium, i.e. R t = R d t . Using Eq. ( 18), we deduce that Eq. ( 11) rewrites as:

d t = θ 1 p t+1 R t+1 + θ 2 φsa t+1 (19)
Then, using the above equilibrium conditions, the budget constraints (2)-(4), Eqs. ( 16) and ( 19), the trade-offs ( 7), ( 8) and (10) write:

θ 1 p t+1 R t+1 -p t (φsR t+2 a t+2 + x t+2 ) = δ (θ 1 p t+1 R t+2 -p t+2 ) sR t a t + θ 2 φsa t+1 + θ 1 p t+1 R t+1 -p t (20)
β (φsR t+2 a t+2 + x t+2 ) = δR t+2 [φsR t+1 a t+1 (1 -θ 2 ) -(1 + φ + φθ 2 s) a t+2 +x t+1 -θ 1 p t+1 + p t+2 R t+2 (21)
β 2 δ (φsR t+2 a t+2 + x t+2 ) = (1 + θ 1 ) p t+2 + (1 + φ + θ 2 φs) R t+2 a t+2 -x t+2 (22)
From Eq. ( 22), we deduce:

x t+2 = δ (1 + θ 1 ) p t+2 + 1 + φ + θ 2 -β 2 /δ φs R t+2 a t+2 β 2 + δ (23)
Let us consider the following assumption:

Assumption 1 δ > δ, with δ ≡ β 2 φs 1+φ+φθ2s . Assumption 1 ensures a strictly non-negative bequests, i.e. x t+2 > 0, by imposing a sufficiently a degree of altruism δ. Substituting the expression of x t defined by Eq. ( 23) into Eqs. ( 20) and ( 21), the intertemporal trade-offs faced by households are summarized by the following equations:

θ 1 p t+1 R t+1 -p t (A 1 R t+2 a t+2 + A 2 p t+2 ) = (θ 1 p t+1 R t+2 -p t+2 ) sR t a t + A 3 a t+1 + θ 1 p t+1 R t+1 -p t (24)
B 1 p t+2 = R t+2 (B 2 R t+1 a t+1 -B 3 p t+1 -B 4 a t+2 ) (25)
with

A 1 ≡ [1 + φ + (1 + θ 2 )φs] /(β 2 + δ) > 0 A 2 ≡ (1 + θ 1 )/(β 2 + δ) > 0 A 3 ≡ θ 2 φs > 0 B 1 ≡ θ 1 + (1 + θ 1 ) β/(β 2 + δ) > 0 B 2 ≡ δ(1 + φ) + (δ -θ 2 β 2 )φs /(β 2 + δ) > 0 B 3 ≡ (β 2 θ 1 -δ)/(β 2 + δ) B 4 ≡ (1 + φ + θ 2 φs) (β + β 2 + δ)/(β 2 + δ) + βφs/(β 2 + δ) > 0
and R t+1 given by Eq. ( 17).

Definition 1 Under Assumption 1, an intertemporal equilibrium with perfect foresight is a sequence (a t , p t ) ∈ R ++ × R + , t = 0, 1, ..., +∞, such that the dynamic system ( 24)-( 25) is satisfied, the borrowing constraint is binding and θ 1

pt+1 pt < R t+1 < 1 θ1 pt+1 pt
, where R t+1 is defined by Eq. ( 17) and a 0 > 0 is given.

In the following, we will use the definition of the equilibrium to analyse the existence of a steady state with a productive real estate bubble, i.e. a bubble associated to a larger level of capital and output per worker.

In accordance with recent contributions that also use OLG model with three-period lived households (Basco, 2014; Raurich and Seegmuller, 2017, 2019), we restrict our attention to the analysis of steady states and do not study the dynamics. In the absence of bubble (p t = p t+1 = p t+2 = 0), the dynamic system collapses to a one-dimensional system and it is easy to see that the capital will converge to its stationary level. When there is a bubble, the dynamics are governed by a fourdimensional system with three non-predetermined variables. Due to its complicated form, a precise analysis of the dynamic properties of this system is beyond the scope of this paper. However, we can conjecture that local indeterminacy might be possible, while we know since the seminal paper of [START_REF] Tirole | Asset bubbles and overlapping generations[END_REF], that models with rational bubbles are characterized by a form of global indeterminacy.

Is the real estate bubble capital enhancing?

From Eq. ( 24), there exist two types of steady states: a bubbleless steady state (i.e. p = 0) and a bubbly steady state (i.e. p > 0). We first characterize the bubbeless steady state, and second the bubbly. Then, we will be able to compare them to show that the bubble may enhance capital.

Bubbleless steady state p = 0

A bubbleless steady state is characterized by p = 0. Using [START_REF] Marcet | Equilibrium asset prices and saving of heterogeneous agents in the presence of incomplete markets and portfolio constraints[END_REF], the gross interest rate R and the capital per labour a are respectively given by:

R = B 4 B 2 and a = α B 2 B 4 1/(1-α) (26)
Note that Assumption 1 ensures a non-negative gross interest rate, i.e. R > 0.

Proposition 1 Let δ ≡ β[(1 + φ + φθ 2 s)(1 + β) + φs(1 + θ 2 β)]/[φs(1 -θ 2 )].
Under Assumption 1, there exist φ and θ 2 > 0 such that there is a unique bubbleless steady state (p, R) with a binding borrowing constraint characterized by p = 0 and R ∈ (0, 1) if δ is sufficiently high such that δ > δ, φ > φ and θ 2 < θ 2 .

Proof. See Appendix 8.2.

To understand that R < 1 is obtained, let us recall that only middle-aged households can invest in physical capital. If households are sufficiently altruistic, then they will leave a high amount of bequest to their middle-aged children, ceteris paribus. As savings are normal good, middle-aged households can save a significant amount in physical capital for a given wage w. This tends to lower the interest rate. If household cannot pledge a huge part of their income (weak θ 2 ), the loan repayment will be weak. For a given wage w, their income after loan repayment will be high, thus middle-aged household will increase their savings in physical capital, ceteris paribus. This tends to lower the interest rate. We finally note that a high enough φ means that labour income is more redistributed to middle-aged. This facilitates a positive borrowing for the young agents, through the redistribution of income from the middle-age to the young.

Bubbly steady state p > 0

A bubbly steady state (p, R, a) must satisfy the following system:

   G(R) = H(R) p = RG(R)a (27) with G(R) ≡ B 2 R -R B 1 + RB 3 (28)
and

H(R) ≡ -(R -θ 1 )A 1 + (1 -θ 1 R)(sR + A 3 ) (R -θ 1 )(A 2 + 1 -θ 1 R) (29)
where R ≡ R(a) is given by Eq. ( 17).

The borrowing constraint is binding if Inequality ( 12) is satisfied. Combining with Eq. ( 7), a binding borrowing constraint implies R < 1 at the steady state. Moreover, Inequality ( 14) must be satisfied at the steady state. Therefore, at the bubbly steady state, R must belong to (θ 1 , 1). To understand why R should be lower than 1, let us recall that the stationary return when old of investing in real estate when young is 1. In contrast, an investment through deposits provides after two periods a return R 2 . Therefore, a household chooses to borrow using deposits to finance real estate up to its credit limit (binding borrowing constraint) only if the return on deposit is lower than the one on real estate. The gross rate R is higher than θ 1 , otherwise, taking into account the binding borrowing constraint and the budget constraint when young, a household could borrow more than the needed amount to finance real estate. This would rule out any equilibrium with a positive and finite price for real estate.

According to the sign of B 3 , two types of bubbly steady states can exist: a steady state with unproductive real estate bubble (R < R or a < a) and a steady state with productive real estate bubble (R > R or a < a). Note that the steady state with productive real estate bubble could occur only if B 3 < 0 and R > -B 1 /B 3 > θ 1 . 6 As a steady state with productive real estate bubble is in accordance with the data, we will restrict our attention to the case for which R > -B 1 /B 3 . To ensure B 3 < 0, we make the following assumption:

Assumption 2 δ > δ, with δ ≡ β 2 θ 1 .
Taking into account that there is a bubbleless steady state, we show the existence of a bubbly steady state with a higher level of production:

Proposition 2 Let θ1 ≡ (1 + φ + φθ 2 s)/(1 + φ + φs).
Under Assumptions 1 and 2, there exists δ > 0 such that a unique steady state with productive real estate bubble (ā, p) characterized by R ∈ (-B 1 /B 3 , R) and p > 0 coexists with the bubbleless steady state if φ is close to but larger than φ, θ 2 < θ 2 , θ 1 < θ1 and δ sufficiently high such that δ > max{ δ, δ, δ}. This means that a > a.

Proof. See Appendix 8.3.

We show that the real estate bubble is productive and the bubbly steady state coexists with the bubbleless one if R < 1. A low interest rate environment means that there is an excess of savings in the economy without bubble, which provides a room for alternative stores of value. At the bubbly steady state, the real estate bubble provides a higher return equal to one. Therefore, this return creates an incentive for households to hold the bubble, since other assets have lower returns and borrowing has a lower cost.

We also observe that, according to Proposition 2, a productive real estate bubble occurs if households are sufficiently altruistic (i.e. high δ), the effects of fundamental and bubbly collaterals are positive but not too important (i.e. θ 2 < θ 2 and θ 1 < θ1 ), and an important share of labor income is distributed to middle-aged agents (i.e. φ high enough).

To understand the main mechanisms explaining that a real estate bubble enhances capital, we rewrite the first and second period budget constraints combined with the borrowing constraint at the bubbly steady state:

c 1 + p (1 -θ 1 /R) = w(1 + φθ 2 /R) (30)
c 2 + 1 + R R (θ 1 p + θ 2 φw) + (1 + φ)a = φw + x (31)
A sufficiently altruistic household want to leave high bequests to their middle-aged children. Investing in the real estate bubble when young and reselling it when old allows her to increase bequests, which pushes up the income of their children at middle-age. This additional income is in particular used to raise the investment in capital, which explains that the real estate bubble is productive.

We deduce from Proposition 2 that a real estate bubble is productive under quite similar conditions on θ 1 and θ 2 . This suggests that fundamental and bubbly collaterals play similar roles on the existence of the real estate bubble, even if the interpretations can be slightly different. By inspection of Eq. ( 30), higher θ 1 and θ 2 increase the role of collateral and allow the household to borrow more when young. Indeed, taking into account the credit constraint, a higher θ 2 generates a positive income effect through the wage, whereas a higher θ 1 implies a lower price of the real estate bubble. Both these effects will promote a higher price of the real estate asset, which increases when old, bequests and therefore investment in capital of the middle-aged households. However, using Eq. ( 31), we observe that at middle-age, a higher θ 1 or θ 2 means a higher debt reimbursed and a larger funding of the borrowing of young agents. Both these effects have a negative impact on capital accumulation and explain that our result does not hold when either θ 1 or θ 2 is too close to one.

We finally note that a high enough value of φ contributes to ensure a significant income at middle-age to finance investments and the reimbursement of the loans contracted while being young.

Of course, the most important ingredient for our explanation of productive real estate bubble lies on altruism. The sufficient conditions we get are based on a sufficiently high degree of altruism. To clarify the crucial role played by altruism to generate a real estate bubble which enhances capital, we analyse now the model without altruism and show that the bubble crowds out capital instead of raising it.

The crucial role of altruism

The model without altruism corresponds to the case where = 0. Its analysis is quite similar to the model with altruism. Therefore, for the sake of brevity, we present the main derivations only.

A household maximizes the utility (1) under the budget constraints ( 2)-( 4) and the borrowing constraint [START_REF] Cocco | Portfolio choice in the presence of housing[END_REF]. Focusing on solutions with a binding borrowing constraint, we get the following optimal conditions:

β 2 c 3,t+2 (p t+2 -θ 1 p t+1 R t+2 ) = 1 c 1,t p t -θ 1 p t+1 R t+1 (32) 1 c 2,t+1 = R t+2 β c 3,t+2 (33)
R d t+2 = R t+2 ( 34
)
θ 1 p t+1 h t+1 + θ 2 φw t+1 = R t+1 d y t ( 35
)
β 2 R d t+1 R t+2 c 1,t < c 3,t+2 (36)
We still need to have θ 1

pt+1 pt < R t+1 < 1 θ1 pt+1 pt . An equilibrium satisfies h t = 1, k t = (1 + φ)a t , d y t = d m t = d t .
Using Eqs. ( 2)-( 4), ( 6) and the trade-offs ( 32) and ( 33), we get:

p t+2 (1 + θ 1 ) + R t+2 a t+2 (1 + φ + φθ 2 s) = βR t+2 [φsR t+1 a t+1 (1 -θ 2 ) -(1 + φ + φθ 2 s) a t+2 -θ 1 p t+1 + p t+2 R t+2 (37)
p t -θ 1 p t+1 R t+1 [p t+2 (1 + θ 1 ) + R t+2 a t+2 (1 + φ + φθ 2 s)] = β 2 (p t+2 -θ 1 p t+1 R t+2 ) sR t a t + θ 2 φsa t+1 + θ 1 p t+1 R t+1 -p t (38)
and the condition for a binding borrowing constraint rewrites:

p t+2 (1 + θ 1 ) + R t+2 a t+2 (1 + φ + φθ 2 s) > β 2 R t+1 R t+2 sR t a t + θ 2 φsa t+1 + θ 1 p t+1 R t+1 -p t (39)
A bubbleless steady state with selfish households (p s , R s ) is characterized by p s = 0. Using (37), the gross interest rate R s is given by:

R s = 1 + β β 1 + φ + θ 2 φs (1 -θ 2 )φs (40)
Moreover, using (39), the borrowing constraint is binding if and only if F s (φ) > 0, with:

F s (φ) ≡ (1 -θ 2 ) 2 sφ 2 -(1 + β)[1 + β + φ(1 + β + θ 2 s(1 + 2β))] (41)
We can easily see that F s (0) < 0 and F s (+∞) = +∞. Therefore, there exists φ s > 0 such that F s (φ) > 0 for all φ > φ s .

Proposition 3 There exists φ s > 0 such that there is a unique bubbleless steady state R s with selfish households with a binding borrowing constraint if φ > φ s .

Let us focus now on a bubbly steady state with selfish households (p s , R s , a s ). Using Eqs. (37) and (38), it satisfies:

p s R s a s = G 0 (R s ) = H 0 (R s ) (42) with G 0 (R) ≡ βφs(1 -θ 2 ) R -R s 1 + θ 1 (1 + β) + θ 1 βR (43)
and

H 0 (R) ≡ β 2 (1 -θ 1 R)(sR + θ 2 φs) -(R -θ 1 )(1 + φ + φθ 2 s) (R -θ 1 )(1 + θ 1 + β 2 -β 2 θ 1 R) (44)
We immediately see that p s > 0 means that G 0 (R s ) > 0, which implies that R s > R s .

Proposition 4 Every bubbly steady state R s with selfish households with a binding borrowing constraint is characterized by R s > R s . This means that without altruism, a stationary real estate bubble can never be productive.

This result confirms that altruism plays a key role for the occurrence of productive real estate bubbles. In the absence of altruism and bequest, middle-aged households have not a high enough income to sustain a higher level of capital when there is a real estate bubble.

Real estate taxation

There is a recurrent debate discussing whether it is better to tax real estate rather than capital to accumulate more productive assets (see [START_REF] Leung | Macroeconomics and housing: A review of the literature[END_REF]). Using our model, we provide an argument showing that a higher tax on real estate diminishes capital. Therefore, it is not clear-cut that one should recommend higher taxes on real estate.

We extend the model with altruism to take into account a tax on real estate. The main idea is to discuss whether a higher tax on real estate does or not generate a higher level of capital. Accordingly, we introduce a proportional tax on real estate τ h (< 1), paid by the consumer when old. Despite the fact that it distorts the relative prices, this tax is used to finance a public good that affects neither the preferences, nor the production. 7With these new ingredients, the budget constraints ( 2)-( 4) write:

c 1,t + p t h t+1 w t + d y t (45) c 2,t+1 + R d t+1 d y t + d m t+1 + k t+2 = φw t+1 + x t+1 (46)
c 3,t+2 + x t+2 = (1 -τ h )p t+2 h t+1 + R t+2 k t+2 + R d t+2 d m t+1 (47)
while the utility is still given by ( 1) with = 1.

There is a government that uses the tax revenues to finance government spendings G t according to the following balanced budget:

G t = τ h p t h t-1 (48)
Since the tax rate τ h is constant, public spendings G t vary to balance the budget at each period of time.

The rest of the model is similar to the one without taxation. Therefore, households maximize utility (1) under the budget constraints (45)-( 47) and the borrowing constraint [START_REF] Cocco | Portfolio choice in the presence of housing[END_REF]. The optimal behaviour of a household is solved in Appendix 8.4. By continuity with respect to the case without taxation, the following inequalities should be satisfied:

θ 1 p t+1 p t < R t+1 < 1 θ 1 (1 -τ h ) p t+1 p t (49)
As in Section 2.5, an equilibrium satisfies h t = 1, k t = (1 + φ)a t and d y t = d m t = d t . Using Eq. (47) and β 2 (φw t+2 + x t+2 ) = δc 3,t+2 , we deduce that:

x t+2 = δ (1 -τ h + θ 1 ) p t+2 + 1 + φ + θ 2 -β 2 /δ φs R t+2 a t+2 β 2 + δ (50)
Under Assumption 1, the bequest is positive whatever the tax rate and the bubble are. Following the same methodology than in Section 2.5, an equilibrium is defined by the following two equations:

p t -θ 1 p t+1 R t+1 (A 1 R t+2 a t+2 + Ã2 p t+2 ) = [(1 -τ h )p t+2 -θ 1 p t+1 R t+2 ] sR t a t + A 3 a t+1 + θ 1 p t+1 R t+1 -p t ( 51
)
p t+2 B1 = R t+2 (B 2 R t+1 a t+1 -B3 p t+1 -B 4 a t+2 ) (52) with Ã2 ≡ 1 -τ h + θ 1 β 2 + δ , B1 ≡ β 1 -τ h + θ 1 β 2 + δ + θ 1 , B3 ≡ β 2 θ 1 -δ(1 -τ h ) β 2 + δ
and A 1 , A 3 , B 2 and B 4 defined as in Section 2.5.

Of course, a bubbleless steady state is not affected by the real estate tax rate τ h . In contrast, a bubbly steady state with taxation (p T , R T , a T ) must satisfy the following system which depends on τ h :

p T R T a T = G T (R T ) = H T (R T ) (53) with G T (R) ≡ B 2 R -R T B1 + R B3 (54)
and

H T (R) ≡ (1 -τ h -θ 1 R)(sR + A 3 ) -(R -θ 1 )A 1 (R -θ 1 )( Ã2 + 1 -τ h -θ 1 R) (55)
The borrowing constraint is binding at the bubbly steady state with taxation if R T < √ 1 -τ h .8 Moreover, Inequality (49) must be satisfied at the steady state, i.e.

θ 1 < R T < (1 -τ h )/θ 1 .
At this stage, we consider that there exists a bubbly steady state with taxation R T ∈ (-B1 / B3 , R T ) whose existence and uniqueness are shown in a similar way than in Proposition 2. 9 We investigate the effect of taxes on such an equilibrium. The role of the tax on real estate τ h is summarized in the following proposition: Proposition 5 Under Assumption 1, consider that there exists a unique bubbly steady state with taxation R T and a binding borrowing constraint. Assuming δ > β/θ 1 , we show that R T is increasing in τ h , meaning that a T is decreasing in τ h .

Proof. See Appendix 8.5.

This proposition shows that an increase of the tax on real estate has a negative effect on investment in productive capital. An increase of τ h reduces the incentive to buy the real estate bubble, which reduces the income when old. Thus, it has a negative effect on bequest distributed to middle-aged agents, who invest less in capital.

This result provides an argument in favour of a reduction of real estate taxation, which may be quite significant, as in France (see Conseil des Prélèvements Obligatoires, 2018).

Concluding remarks

In this paper, we present a three-period OLG model with capital accumulation, debt, real estate and bequests. Our model presents three features. First, households realise different investments at different period of life. They borrow when young to buy a real estate asset for their old age and save when middle-aged to transfer resources to their old age. Second, households face a binding borrowing constraint. Because of credit market imperfections, they must pledge their wages and their real estate asset as collaterals. Third, households behave altruistically by leaving bequests to their children.

We exhibit the existence of a steady state with a real estate bubble accompanied by a boom in physical capital and aggregate output. This result mainly relies on the existence of altruism in our model. Without altruism, middle-aged households have not a high enough income to sustain a higher level of capital in the presence of a real estate bubble. There is no room for such a bubble. Moreover, we show that a productive real estate bubble exists whatever the type of collateral (fundamental or bubbly collateral) required by lenders in the credit market.

We further analyse the effects of real estate taxation on capital accumulation. We show that an increase of real estate taxation dampens capital accumulation, which is harmful for the real economy. Therefore, our result provides an argument against the implementation of taxation on unproductive assets as real estate.

Appendix

Households' behaviour

We maximize the Lagrangian function:

lnc 1,t + βlnc 2,t+1 + β 2 lnc 3,t+2 + δln(φw t+2 + x t+2 ) +λ 1,t (w t + d y t -c 1,t -p t h t+1 ) + λ 2,t+1 φw t+1 + x t+1 -c 2,t+1 -R d t+1 d y t -k t+2 -d m t+1 +λ 3,t+2 p t+2 h t+1 + R d t+2 d m t+1 + R t+2 k t+2 -c 3,t+2 -x t+2 +λ 4,t+1 θ 1 p t+1 h t+1 + θ 2 φw t+1 -R d t+1 d y t
with respect to (c 1,t , c 2,t+1 , c 3,t+2 , k t+2 , d y t , d m t+1 , h t+1 , x t+2 , λ 1,t , λ 2,t+1 , λ 3,t+2 , λ 4,t+1 ).

First-order conditions are given by:

c 1,t : 1 c 1,t = λ 1,t (56)
c 2,t+1 : β c 2,t+1 = λ 2,t+1 (57)
c 3,t+1 :

β 2 c 3,t+2 = λ 3,t+2 (58)
h t+1 : p t λ 1,t -p t+2 λ 3,t+2 -θ 1 p t+1 λ 4,t+1 = 0 (59) From Eqs. ( 61) and ( 63), one has:

x t+2 : λ 3,t+2 - δ φw t+2 + x t+2 = 0 (60) d y t : λ 1,t = R d t+1 (λ 2,t+1 + λ 4,t+1 ) (61)
d m t+1 : λ 2,t+1 = R d t+2 λ 3,t+2 (62)
k t+2 : λ 2,t+1 = R t+2 λ 3,t+2 (63)
λ 4,t+1 = λ 1,t R d t+1 -R t+2 λ 3,t+2 (65)
We restrict our attention to the case of a binding borrowing constraint to be able to discuss our results according to the importance of the different types of collateral. Using (64) and the Kuhn and Tucker conditions, this is ensured by λ 4,t+1 > 0. Substituting Eqs. ( 56)-( 58) and (65) into Eqs. ( 59)-( 63), one has :

β 2 c 3,t+2 (p t+2 -θ 1 p t+1 R t+2 ) = 1 c 1,t p t -θ 1 p t+1 R t+1 (66)
β 2 c 3,t+2 = δ φw t+2 + x t+2 (67) 1 c 2,t+1 = R d t+2 β c 3,t+2 (68)
θ 1 p t+1 h t+1 + θ 2 φw t+1 = R d t+1 d t (69)

Proof of Proposition 1

One can easily show that R < 1 if and only if δ > δ. The borrowing constraint is binding if

c 3 > β 2 R 2 c 1 .
This inequality is equivalent to:

A 1 > R (sR + A 3 ) (70)
Since R < 1 under δ > δ, inequality (70) is satisfied if

A 1 > sR + A 3 (71)
We can write A 1 and sR + A 3 as continuous functions of φ and θ 2 :

A 1 = 1 + φ [1 + (1 + θ 2 )s] β 2 + δ ≡ LHS(φ, θ 2 ) (72) sR + A 3 = s β + β 2 + δ + φ[(1 + θ 2 s)(β + β 2 + δ) + βs] δ + φ[(1 + s)δ -β 2 θ 2 s] + φθ 2 s ≡ RHS(φ, θ 2 ) (73) Note that LHS(φ, 0) = 1 + φ [1 + s] β 2 + δ (74) RHS(φ, 0) = s β + β 2 + δ + φ[β(1 + s) + β 2 + δ] δ + φ(1 + s)δ (75)
For θ 2 = 0, Inequality (70) is satisfied if LHS(φ, 0) > RHS(φ, 0). Note that LHS(φ, 0) is an increasing function of φ and RHS(φ, 0) a decreasing function of φ. We have LHS(+∞, 0) > RHS(+∞, 0) and LHS(1, 0) < RHS(1, 0) if δ is sufficiently high. Therefore, it exists a threshold φ 0 > 1 such that for φ > φ 0 , LHS(φ, 0) > RHS(φ, 0). Note that LHS(φ, θ 2 ) and RHS(φ, θ 2 ) are both increasing functions of θ 2 . Moreover, for δ high enough, LHS(+∞, 1) < RHS(+∞, 1). By continuity, there exist thresholds θ 2 ∈ (0, 1) and φ > 1 such that for θ 2 < θ 2 and φ > φ, LHS(φ, θ 2 ) > RHS(φ, θ 2 ). Therefore, if δ is sufficiently high such that δ > δ, φ > φ and θ 2 < θ 2 , Proposition 1 follows.

Proof of Proposition 2

A bubbly steady state R is a solution of G(R) = H(R), where G(R) and H(R) satisfy ( 28)- [START_REF] Mera | Asia's Financial Crisis and the Role of Real Estate[END_REF].

We start by determining the admissible range of values for R. We have already seen that θ 1 < R < 1. We remind that we restrict our attention to the existence of a productive bubble (i.e. R < R). As p = RG(R)a, B 1 + RB 3 must be negative to get a positive bubble.

As

B 1 = θ 1 + (1 + θ 1 ) β/(β 2 + δ) > 0, B 1 + RB 3 < 0 implies R > -B 1 /B 3 with B 3 = (β 2 θ 1 -δ)/(β 2 + δ) < 0.
Note that under Assumption 2 (δ > δ), B 3 < 0 and -B 1 /B 3 > θ 1 . Then, the possible admissible range of values for productive bubbles is R ∈ (-B 1 /B 3 , R), with R < 1.

To prove the existence of a stationary solution R ∈ (-B 1 /B 3 , R), we use the continuity of G(R) and H(R). Note that G(R) has a vertical asymptote at R = -B 1 /B 3 and is a decreasing and continuous function of R when R increases to R. H(R) has two vertical asymptotes at R = θ 1 < -B 1 /B 3 and R = (1 + A 2 )/θ 1 > 1. From Eqs. ( 28) and ( 29), we determine the boundary values of G(R) and H(R):

lim R→-B1/B3 G (R) = +∞ lim R→R G (R) = 0 lim R→-B1/B3 H (R) = (B 1 /B 3 + θ 1 )A 1 + (1 + θ 1 B 1 /B 3 )(-sB 1 /B 3 + A 3 ) (-B 1 /B 3 -θ 1 )(A 2 + 1 + θ 1 B 1 /B 3 ) lim R→R H (R) = -(R -θ 1 )A 1 + (1 -θ 1 R)(sR + A 3 ) (R -θ 1 )(A 2 + 1 -θ 1 R) We have lim R→-B1/B3 H (R) < lim R→-B1/B3 G (R). The existence of a steady state R ∈ (-B 1 /B 3 , R), solving G(R) = H(R), is ensured by 0 < H(R), namely: A 1 < 1 -θ 1 R R -θ 1 (sR + A 3 ) (76) Note that (1 -θ 1 R)/(R -θ 1 ) > 1 if
Note that LHS(δ, θ 1 ) and RHS(δ) are decreasing functions of δ and we have that:

lim δ→+∞ LHS (δ, θ 1 ) = θ 1 lim δ→+∞ RHS (δ) = 1 + φ + φθ 2 s 1 + φ + φs < 1
We deduce that lim δ→+∞ LHS (δ, θ 1 ) < lim δ→+∞ RHS (δ) if and only if:

θ 1 < 1 + φ + φθ 2 s 1 + φ + φs ≡ θ1
If θ 1 < θ1 , there exists a threshold δ > 0 such that for δ > δ, LHS(δ, θ 1 ) < RHS(δ). Therefore, if θ 1 < θ1 and δ > δ, the set (-B 1 /B 3 , R) is nonempty.

To conclude, if φ is close but larger than φ, θ 2 < θ 2 , θ 1 < θ1 and δ sufficiently high such that δ > max{ δ, δ, δ}, there exists a steady state R ∈ (-B 1 /B 3 , R) which coexists with the bubbleless one R.

To prove uniqueness, we observe that the equation H(R) = G(R) is equivalent to a polynomial of degree 3, i.e. has at most three solutions. Since lim R→(1+A2)/θ1 H (R) = -∞ and lim R→(1+A2)/θ1 G (R) is finite, we have lim R→(1+A2)/θ1 H (R) < lim R→(1+A2)/θ1 G (R). We have established that lim R→R H (R) > lim R→R G (R). This means that an odd number of solutions to the equation H(R) = G(R) belong to (R, (1 + A 2 )/θ 1 ). Since we also have lim R→-B1/B3 G (R) > lim R→-B1/B3 H (R), there are an odd number of solutions to H(R) = G(R) that belong (-B 1 /B 3 , R).

Since there are at most three solutions, the solution R in this last interval is unique.

Households' behaviour in the model with taxation

We maximize the Lagrangian function:

lnc 1,
From Eq. (83), we get R t+2 = R d t+2 . From Eqs. (82) and (83), one has:

λ 4,t+1 = λ 1,t R d t+1 -R d t+2 λ 3,t+2 (84)
We restrict our attention on the case where λ 4,t+1 > 0, i.e. λ 1,t > R d t+1 R d t+2 λ 3,t+2 . Substituting Eq. (79) into Eqs. (80), ( 81) and (83), we deduce:

β 2 c 3,t+2 [(1 -τ h )p t+2 -θ 1 p t+1 R t+2 ] = 1 c 1,t p t -θ 1 p t+1 R t+1 (85) 1 c 2,t+1 = R t+2 β c 3,t+2 (86)
R d t+2 = R t+2 (87)
β 2 c 3,t+2 = δ φw t+2 + x t+2
(88)

θ 1 p t+1 h t+1 + θ 2 φw t+1 = R t+1 d y t ( 89
)
β 2 R t+1 R t+2 c 1,t < c 3,t+2 (90)
x t+2 > 0 (91)

Proof of Proposition 5

The existence of the steady state is obtained because G T (R) > H T (R) when R tends to -B1 / B3 and G T (R) < H T (R) when R tends to R T . Uniqueness allows to deduce that at the steady state, we have G T (R) < H T (R). Using Eq. ( 53), we have:

dR T d(1 -τ h ) = ∂G T (R T ) ∂(1-τ h ) -∂H T (R T ) ∂(1-τ h ) H T (R T ) -G T (R T ) (92)
We start by computing ∂G T (R T )/∂(1 -τ h ). Note that B 2 and B 4 do not depend on τ h , while we have:

∂( B1 + R B3 ) ∂(1 -τ h ) = β -δR β 2 + δ < 0
because R > θ 1 and δ > β/θ 1 . We deduce that:

∂G T (R T ) ∂(1 -τ h ) = - G T (R T ) B1 + R T B3 β -δR T β 2 + δ < 0 (93)
because B1 + R T B3 < 0. Using Eq.( 55), H T (R) can be rewritten as follows:

H T (R) = β 2 + δ R -θ 1 (1 -τ h -θ 1 R)(sR + A 3 ) -(R -θ 1 )A 1 (1 -τ h )(1 + β 2 + δ) + θ 1 [1 -(β 2 + δ)R] (94)
Obviously, ∂H T (R T )/∂(1 -τ h ) has the sign of its numerator, i.e.

(sR

T + A 3 )θ 1 (1 + R T ) + (1 + β 2 + δ)A 1 R T -θ 1 > 0
Using Eq. (92), we deduce that dR T /d(1 -τ h ) < 0. This means that R T increases with τ h , whereas a T decreases with τ h .

λ 4 ,

 4 t+1 θ 1 p t+1 h t+1 + θ 2 φw t+1 -R d t+1 d t = 0 (64)From Eqs. (62)-(63), we get : R t+2 = R d t+2 . Physical capital and deposit are perfect substitutes.

  and only if R < 1. Using the proof of Proposition 1, Inequality (76) is in accordance with Inequality (71) and satisfied by continuity argument if φ close but larger than φ and θ 2 ∈ (0, θ 2 ), because it ensures that A 1 is larger but close to sR + A 3 .We ensure now that the set (-B 1 /B 3 , R) is non-empty. The set (-B 1 /B 3 , R) is non-empty if and only if -B 1 /B 3 < R with:

	-	B 1 B 3	=	δθ 1 + β + θ 1 β(1 + β) δ -β 2 θ 1	≡ LHS(δ, θ 1 )	(77)

R = (1 + φ + φθ 2 s)(δ + β + β 2 ) + βφs δ(1 + φ + φs) -θ 2 β 2 φs ≡ RHS(δ)

  t + βlnc 2,t+1 + β 2 lnc 3,t+2 + δln[φw t+2 + x t+2 ] +λ 1,t (w t + d y t -c 1,t -p t h t+1 ) + λ 2,t+1 φw t+1 + x t+1 -c 2,t+1 -R d t+1 d y t -k t+2 -d m t+1 +λ 3,t+2 (1 -τ h )p t+2 h t+1 + R d t+2 d m t+1 + R t+2 k t+2 -c 3,t+2 -x t+2 +λ 4,t+1 θ 1 p t+1 h t+1 + θ 2 φw t+1 -R d t+1 d y twith respect to (c 1,t , c 2,t+1 , c 3,t+2 , k t+2 , d y t , d m t+1 , h t+1 , x t+2 , λ 1,t , λ 2,t+1 , λ 3,t+2 , λ 4,t+1

  t λ 1,t = (1 -τ h )p t+2 λ 3,t+2 + θ 1 p t+1 λ 4,t+1 R d t+2 λ 3,t+2 , λ 2,t+1 = R t+2 λ 3,t+2

								). First-order
	conditions are given by:
	1 c 1,t	= λ 1,t ,	β c 2,t+1	= λ 2,t+1 ,	β 2 c 3,t+2	= λ 3,t+2	(79)
								(80)
	λ 3,t+2 =	δ φw t+2 + x t+2			(81)
	λ 1,t = R d t+1 (λ 2,t+1 + λ 4,t+1 )			(82)
	λ 2,t+1 =

p

In their papers,[START_REF] Kamihigashi | The spirit of capitalism, stock market bubbles and output fluctuations[END_REF] and more recently Miao et al. (2017) provide some historical examples revealing that the bursting of a real estate bubble is often followed by a recession.

On the one hand, empirical evidence confirms that real estate collateral plays a significant role during crisis[START_REF] Mera | Asia's Financial Crisis and the Role of Real Estate[END_REF]. Moreover, macroeconomics model dealing with financial frictions consider a real estate collateral constraint along the lines of[START_REF] Kiyotaki | Credit cycles[END_REF] andIacoviello (2005). On the other hand, some studies argue that the debt limit also depends on household income(Marcet and Singleton, 1991;[START_REF] Ludvigson | Consumption and credit: A model of time-varying liquidity constraints[END_REF].

Martin and Ventura (2011, 2016) use a quite similar constraint except that the fundamental collateral corresponds to entrepreneur's income.

We assume a full capital depreciation within a period.

In OGL models, Arce and Lopez-Salido (2011) and Basco (2014) also study housing bubbles. Nevertheless, they model housing bubble as a shortage of assets in the economy, and not as a component of the housing price.

According to[START_REF] Martin | Economic growth with bubbles[END_REF], a stationary bubble means G(R) > 0.

In our paper, we consider wasteful public expenditures, and thus abstract from productivity-enhancing and utility-enhancing public expenditures. By not modelling productive government spendings, our results are thus not triggered by the existence of some positive productive externalities. As regards utility-enhancing public expenditures, we can note that an additively separable utility function would not change our results.

Using Appendix 8.4, the borrowing constraint is binding combining Inequality (90) and Eq. (85).

Of course, by continuity, this is obvious if the tax rate is low enough.

This work has benefited from the financial support of the French National Research Agency (ANR-15-CE33-0001-01).


Théo Ladune

Pierrick Philippe

Wassim Hamidouche

Lu Zhang

Olivier D Éforges

BINARY PROBABILITY MODEL FOR LEARNING BASED IMAGE COMPRESSION

Keywords: Image Coding, Autoencoder, Entropy Coding, Convolutional Neural Network

In this paper, we propose to enhance learned image compression systems with a richer probability model for the latent variables. Previous works model the latents with a Gaussian or a Laplace distribution. Inspired by binary arithmetic coding, we propose to signal the latents with three binary values and one integer, with different probability models.

A relaxation method is designed to perform gradientbased training. The richer probability model results in a better entropy coding leading to lower rate. Experiments under the Challenge on Learned Image Compression (CLIC) test conditions demonstrate that this method achieves 18 % rate saving compared to Gaussian or Laplace models.

INTRODUCTION

Data compression can be summarized in three main steps. First, the input signal is encoded into more compact variables called latents. Then, the latents are transmitted with a coding method achieving a rate near to the Shannon entropy. Lastly, the input signal is decoded from the latents. As a real number has an infinite information quantity (i.e. an infinite number of bits), lossy coding methods only work with finite set of values. To address this issue, latents are quantized, introducing distortion on both the latents and the reconstructed signal.

Lossy image compression can thus be expressed as an optimization problem: jointly minimizing the distortion and the rate (i.e. information in the latents). Traditional coding approaches such as JPEG or BPG (HEVC-based image compression) [START_REF] Wallace | The jpeg still picture compression standard[END_REF]2] typically solve this problem using linear predictions and transforms. Deep neural networks can learn complex non-linear functions, making them well suited to reach better optimum and coding efficiency. However, the discrete nature of the data sent from the encoder to the decoder makes the objective function non-differentiable and prevents optimizing end-to-end systems with gradient-based methods.

In [START_REF] Ballé | End-to-end optimized image compression[END_REF], authors suggest to replace quantization with additive noise and propose an interpolation of the rate function.

A different quantization approximation is presented in [START_REF] Theis | Lossy image compression with compressive autoencoders[END_REF]. These works show promising results, outperforming the JPEG standard.

Entropy coding requires an estimate of the latents probability density function (PDF). Whereas previous works use a fixed-PDF model, Ballé et al. introduce hyperpriors in [START_REF] Ballé | Variational image compression with a scale hyperprior[END_REF], consisting in side-information conditioning each latent PDF. This more accurate probability model brings important performance gains. Minnen et al. and Lee et al. [START_REF] Minnen | Joint autoregressive and hierarchical priors for learned image compression[END_REF][START_REF] Lee | Context-adaptive entropy model for end-toend optimized image compression[END_REF] add an autoregressive model (ARM) to infer PDF parameters from previously sent values. However, such systems lead to a prohibitive decoding time due to the sequential nature of the ARM which is not suited for GPU processing.

In 2019, the Challenge on Learned Image Compression (CLIC) [START_REF]Workshop and Challenge on Learned Image Compression[END_REF] was held at the Conference on Computer Vision and Pattern Recognition (CVPR), providing a common evaluation framework to the learned image compression community. Proposed end-to-end systems [START_REF] Zhou | End-to-end optimized image compression with attention mechanism[END_REF][START_REF] Wen | Variational autoencoder based image compression with pyramidal features and context entropy model[END_REF] composed of a hyperprior and an ARM outperformed BPG [2].

Improvements of the latents probability model are the main reason behind the successive performance gains. In this paper, we propose a more accurate estimate of the latents PDF widely inspired by the HEVC binarization process [START_REF] Sullivan | Overview of the high efficiency video coding (hevc) standard[END_REF]. Based upon Minnen's work [START_REF] Minnen | Joint autoregressive and hierarchical priors for learned image compression[END_REF], we present a new relaxation method for a discrete rate function. This allows to leverage the richer probability model providing either better performance with the same complexity or similar performance with a lightweight coding system.

PROPOSED METHOD

Framework description

The work carried out in this paper is based upon Ballé and Minnen's work [START_REF] Ballé | End-to-end optimized image compression[END_REF][START_REF] Ballé | Variational image compression with a scale hyperprior[END_REF][START_REF] Minnen | Joint autoregressive and hierarchical priors for learned image compression[END_REF]. Their framework for training endto-end lossy compression system is explained in this section. The architecture is the one described in [START_REF] Minnen | Joint autoregressive and hierarchical priors for learned image compression[END_REF]. Fig. 1 illustrates the coding scheme which can be summarized as:

1. Encoding the input image x into latents y = g a (x; θ e ); 2. Encoding the hyperprior z = h a (y; θ he ); The set of neural network parameters {θ e , θ d , θ he , θ hd } is learnt by minimizing a rate-distortion trade-off

L(λ) = D(x, x) + λ (R(ŷ) + R(ẑ)) .
In this work, the distortion is computed through the meansquared error

D(x, x) = E x∼px ||x -x|| 2 .
Latents ŷ and the hyperprior ẑ are encoded with arithmetic coding, a lossless coding method achieving a rate near to Shannon entropy

R(ŷ) = E ŷ∼m [L(ŷ; P ŷ)] = E ŷ∼m [-log 2 P ŷ(ŷ)],
where m denotes the distribution of latents (which is unknown) and L is the code length computed thanks to the probability model P ŷ. This can be re-written as [START_REF] Lee | Context-adaptive entropy model for end-toend optimized image compression[END_REF]:

R(ŷ) = H(m) + D KL (m || P ŷ),
where D KL denotes the Kullback-Leibler divergence. Thus, minimizing the rate implies to jointly lower the entropy H(m) of ŷ and properly match the distribution m with the probability model P ŷ. This also holds for rate of ẑ.

Training neural networks relies on gradient-based algorithms, requiring all operations to be differentiable. Because quantization derivative is null almost everywhere, it is modeled as an additive uniform noise during training [START_REF] Ballé | End-to-end optimized image compression[END_REF]

ỹ = y + u ⇒ p ỹ = p y * p u , u ∼ U(-1 2 , 1 2 ),
where p denotes probability distribution. Continuous interpolation L(ỹ; p ỹ) = -log 2 p ỹ(ỹ) of the code length function is used as a proxy to optimize discrete L(ŷ; P ŷ). The same goes for ẑ and the loss function becomes

L(λ) = E x∼px ||x -x|| 2 + λ( L(ỹ; p ỹ) + L(z; p z) . (1)
The hyperprior distribution p z is estimated through a fixed model described in [START_REF] Ballé | Variational image compression with a scale hyperprior[END_REF]. Each latent y i is coded independently and their distribution p yi ∼ N (µ i , σ i ) is decoded from the hyperprior

L(ỹ; p ỹ) = i L(ỹ i , p ỹi ) = i -log 2 (p yi * p u ) (ỹ i ) = i -log 2 ỹi+ 1 2 ỹi- 1 2 N (u; µ i , σ i ) du. (2)
In this paper, we enhance the probability model p yi in order to improve the entropy coding efficiency. As in traditional video coding, latents are transmitted in a binary version, allowing a more accurate model p yi .

Binary probability model

For the sake of clarity, latents index is omitted i.e. y stands for any y i . The purpose of this work is to relax assumptions on p y . To do so, each latent is represented with three binary values and one integer with separate probability model. First, the expectation µ is decoded from the hyperprior and used to center y before quantization: ŷ = Q(y -µ). Each ŷ is then signaled as described in Table 1.

ŷ Elements transmitted Code length L bin G 0 G 1 S E 0 0 L G0 ± 1 1 0 ±1 L G0 + L G1 + L S ± k 1 1 ±1 k L G0 + L G1 + L S + L E Table 1: G 0 (respectively G 1
) stands for greater than zero (respectively one), S for sign and E for explicit.

Flags G 0 and G 1 are transmitted using an entropy coding method, their code length is estimated as

L G X = -log 2 P G X if G X = 1, -log 2 (1 -P G X ) otherwise X = {0, 1} .
Probabilities P G0 and P G1 are decoded from the hyperprior ẑ.

The sign flag is assumed equiprobable costing L S = 1 bit. A latent |ŷ| ≥ 2 is explicitly transmitted with a code length estimated as

L E (k) = -log 2 P ŷ (|ŷ| = k |ŷ| > 1). (3)
Here, p y is modelled as a centered Laplace distribution with σ decoded from the hyperprior. Equation (3) becomes The total code length L bin is obtained by adding up all transmitted elements (cf. Table 1). All ŷ ∈ {-1, 0, 1} are no longer constrained to a pre-determined distribution as P ŷ can represent any symmetrical probability distribution in this interval. The entropy coding of each latent y requires the set {µ, σ, P G0 , P G1 }. Hence, the decoded hyperprior ψ has four features per ŷ: in Fig. 1 H = 4B.

L E (k) = -log 2 2 k+0.5 k-0.5 L(u; 0, σ) du 1 - 1.5 -1.5 L(u; 0, σ) du . ( 4
)

Relaxed rate

The previous section proposes a richer representation of P ŷ . During training, discrete ŷ is replaced by a continuous ỹ, requiring the interpolation of the code length function L. As no hypothesis is made on p y , eq. ( 2) can not be used directly. A new interpolation Lbin is introduced as a weighted sum of the two nearest integer rates:

Lbin (ỹ) = Γ(|ỹ|)L bin ( ỹ ) + (1 -Γ(|ỹ|)) L bin ( ỹ + 1),
where • denotes the floor function. Γ(ỹ) is a weighting function defined with linear segments and depicted in Fig. 2. The main design constraint on the weighting function Γ is to ensure that Lbin (k) = L bin (k) for all integers k to make training and inference metrics coherent. Because sending ŷ = 0 requires only one element (G 0 ), the optimization process results in zeros being the most present value. The flat zone in [0, 1 2 ] is used to make the optimization focus more on the cost of zeros. In [1, +∞] interval, Γ is a simple linear weighting based on the distance to the nearest integer. With the relaxed rate, the loss function becomes:

L(λ) = E x∼px [||x -x|| 2 -λ( Lbin (ỹ) + L(z; p z))].

EXPERIMENTAL RESULTS

Performance on CLIC low-rate task

The proposed method is evaluated on the CLIC 2019 lowrate task [START_REF]Workshop and Challenge on Learned Image Compression[END_REF]. The objective is to achieve the highest PSNR at 0.15 bit per pixel (bpp). For all experiments, the training set is constructed by concatenating the CLIC and DIV2K [START_REF] Agustsson | Ntire 2017 challenge on single image super-resolution: Dataset and study[END_REF] datasets. The 3 000 pictures of these datasets are transformed into non-overlapping 256 × 256 crops. Minibatches of size 8 and Adam algorithm with a learning rate of 10 -4 are used.

The training lasts 80 epochs and the learning rate is divided by 5 at the 50 th and 70 th epoch.

The network described in Fig. 1 is used to evaluate three probability models: Gaussian, Laplace and binary. For all experiments, B = 76 ŷ features and C = 24 ẑ features are transmitted. Transforms g a , g s and h a always have the same complexity. The transform h s is slightly modified due to the number of features (denoted as H in Fig. 1) needed to parameterize latents distribution (H = 2B for Gaussian and Laplace, H = 4B for binary model). Hence, different performance levels are entirely explained by the probability model. The models are evaluated with lightweight (M = 64) and standard (M = 192) configurations.

The rate is estimated by the latents entropy. Performance at 0.15 bpp is obtained by training systems with a λ setting a working point close to the target rate. During inference, the quantization step can be slightly deviated from 1 to plot rate distortion curve around the training point. This enables to accurately estimate the rate at 0.15 bpp and to compute BD rates [START_REF] Bjontegaard | Calculation of average psnr differences between rd-curves[END_REF] by comparing RD curves in [0.13, 0.17] bpp interval. BD rate represents the rate difference necessary to obtain the same PSNR quality between two systems.

Figure 4 and Table 3 sum up results on CLIC 2019 validation and test sets, composed of 102 and 330 various resolution images . Gaussian systems are re-implementations of Minnen et al. [START_REF] Minnen | Joint autoregressive and hierarchical priors for learned image compression[END_REF] without the autoregressive component and are used as a baseline. Laplacian is added as [START_REF] Zhou | Variational autoencoder for low bit-rate image compression[END_REF] argues that it slightly improves performances. BPG is also added as it is the image version of HEVC, the state-of-the-art video coding standard.

The proposed method shows significant rate savings in all configurations, up to 18.3 %. This proves the benefits of a richer PDF model to perform a more efficient entropy coding. Binary probability model brings 9.1 % rate saving for standard systems, achieving results competitive with BPG. Performance improvements are greater with lightweight systems. It may be because they have less powerful transforms g a and g s . Indeed, relaxing the constraints p y makes the system focus more on creating useful latents instead of matching a given PDF. This holds for standard systems to a lesser extent. Finally, it is worth noting that the binary model lightweight system can reach the performance of the standard Gaussian system with 10 times less parameters.

Illustration

Figure 5 depicts the processing of an image by the binary model system. On the left side, feature map ŷ65 is the costliest feature map (around 7 % of the rate). Many pixels are greater than one, resulting in high probabilities for P G0 and P G1 . As most of the values have important dynamic and need explicit sending, the scale parameter σ takes a wide range of values. On the right side, feature map ŷ51 is very sparse and consists mostly in details, representing only 2 % of the rate. Entirely null areas, as the sky, are well captured by the hyperprior, with a very low probability of being greater than zero. This allows to code them with fewer bits. ) is the probability for a pixel to be greater than 0 (respectively 1). σ is the scale parameter used for explicit latents sending.

Systems

Fig. 1 : 6 .

 16 Fig. 1: Network architecture. Rounded arrows denote GDN [3] and squared arrows LeakyReLU. Convolution parameters are: filters number × kernel height × width / stride. Upscaling convolutions are transposed convolutions.

Fig. 2 :

 2 Fig. 2: The weighting function Γ.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Latents probability models performances.