Rational Bubbles and Beneficial Public Debt

Rational Bubbles and Beneﬁcial Public Debt: Arbitrage Limits in Dynastic Economies∗

David Domeij†and Tore Ellingsen‡

March 18, 2016

Abstract

We develop a dynastic general equilibrium model that admits bubbles and Ponzi schemes under rational expectations. Because current rents on average grow more slowly than aggregate output, while future rents are stochastic and imperfectly pledgeable, all agents are eventually liquidity constrained. Hence, although short-selling is costless, arbitrage does not eliminate non-fundamental asset values. In the most comprehensive version of our framework, precautionary saving is caused by imperfectly insurable labor income and entrepreneurial rents. Agents can save in physical capital, claims on intangible capital, and government bonds as well as in private bubbles. Calibrating the model to US data, we argue that public debt is a backed Ponzi scheme; it could be settled in ﬁnite time, but it could also constitute a considerable share of output indeﬁnitely. However, paying down public debt would entail large welfare losses. For reasons of social insurance, it is better to harbor non-fundamental asset values in the form of public debt than in the form of private bubbles. Governments that can sustain more public debt should therefore tax private assets that are otherwise prone to bubbles.

JEL classiﬁcation: E21, E31 Keywords: Bubbles, Incomplete Markets, Fiscal Policy

∗We thank Klaus Adam, Tobias Broer, Ferre de Graeve, Denis Gromb, John Hassler, Paul Klein, Per Krusell, Espen Moen, John Moore, Lars Ljungqvist, Jose-V´ ´ıctor R´ıos-Rull, Asbjørn Rødseth, Jose´ Scheinkman, Per Stromberg,¨ Lars E.O. Svensson, Jean Tirole, Jaume Ventura, and Michael Woodford for valuable discussions, several anonymous referees for objecting to earlier versions, and the Ragnar Soderberg¨ Foundation (Ellingsen) and the Jan Wallander and Tom Hedelius Foundation (Domeij) for ﬁnancial support. †Stockholm School of Economics, [email protected] ‡Stockholm School of Economics, Norwegian School of Economics, [email protected]. 1 Introduction

If people have rational expectations, can asset prices have bubbles? In a growing economy, will it be possible for governments to earn a recurrent “free lunch” by raising their debts in tandem with output? A literature on public debt and asset prices in incomplete markets demonstrates that there are conditions under which the answers to both questions are affirmative; see in particular Allais (1947), Samuelson (1958), Dia- mond (1965), Bewley (1980,1983), and Tirole (1985).1 Bubbles and Ponzi-schemes are logically compatible with rational behavior and beliefs. However, for the last couple of decades – after the objections of, among others, Barro (1974), Abel et al. (1989), and Santos and Woodford (1997) – mainstream profes- sional opinion has been that the necessary conditions for rational bubbles and Ponzi- schemes are unlikely to be fulfilled in practice. Thus, the theory of asset prices in general and fiscal and monetary theory in particular must rest on other foundations. Santos and Woodford (1997, page 48) summarize their insights as follows:

These results suggest that known examples of pricing bubbles depend upon rather special circumstances. In consequence, familiar examples (such as the overlapping generations model of Samuelson (1958) or the Bewley (1980) model) seem to be quite fragile as potential foundations for monetary theory. For when these models are extended to allow for trading in additional assets that are sufﬁciently productive (in particular, capital earning returns satisfying the criterion of Abel et al.), or to include an inﬁnitely lived house- hold (or Barro “dynasty”) that is able to borrow against its future endowment and that owns a fraction of aggregate wealth, pricing bubbles – and hence monetary equilibria – can be excluded under quite general assumptions.

Likewise, LeRoy (2004, page 801) reluctantly concludes:

Within the neoclassical paradigm there is no obvious way to derail the chain of reasoning that excludes bubbles.

It seems we must accept either that bubbles do not exist or, like LeRoy, continue to believe that bubbles exist in reality, but suspend our faith in models that rest on purely neoclassical assumptions.

1Other notable early contributions to this literature include Wallace (1980), Blanchard and Watson (1982), and Weil (1987); for further references, see Blanchard and Fischer (1989, Chapter 5) and Farhi and Tirole (2012).

2 We argue that this judgment is premature. Barro (1974) demonstrates that altruism can rectify the market incompleteness in Diamond (1965), but does not preclude the possibility that asset markets in such dynastic economies could be incomplete for other reasons. The empirical finding of Abel et al. (1989) that the economy is dynamically efficient – i.e., average dividend and interest payments exceed average investment – only implies that there cannot be bubbles on assets whose return is as high as the average return on capital.2 While this finding technically rejects all models of rational bubbles that only admit a single interest rate, such as Samuelson (1958), Diamond (1965), and Tirole (1985), there are closely similar models with multiple interest rates in which there can be bubbles on low-yielding assets as well as Ponzi public debt. This was noted in various contexts already by Woodford (1990), Bertocchi (1991), and Blanchard and Weil (2001),3 and is a crucial justification for more recent theories of rational bubbles, such as Farhi and Tirole (2012). However, the theoretical arguments of Santos and Woodford (1997) (henceforth SW) are apparently stronger. They say that dynastically oriented agents will be able to arbitrage away all bubbles in economies with a plausibly rich asset structure. The goal of this paper is to dispute the relevance of SW’s assumptions. In a nutshell, we claim that incomplete markets create natural limits to arbitrage; any dynasty that tries to make an arbitrage profit from a rational bubble is bound to go broke and be forced to close the position before any profit can be harvested.4 Specifically, we present a quantitative model that admits bubbles and Ponzi-schemes without unrealistic restrictions on the set of tradable assets. We pay particular attention to calibrations in which non- fundamental asset values are large. In these calibrations, public debt corresponding to current US levels is both sustainable and costless. The debt is also backed, in the sense that it could be entirely repaid in finite time. Yet, along a balanced growth path public debt constitutes a constant and large fraction of output, and debt service never requires primary surpluses. If debt were to be further increased until the equilibrium real interest rate on government bonds exceeded the economy’s rate of growth, ratio-

2Geerolf (2013) argues that aggregate data do not admit Abel et al’s (1989) conclusion that US or the other advanced economies are dynamically efﬁcient. However, if we conﬁne attention to the business sector, it seems quite clear that weighted average returns to stocks and bonds exceed the economy’s growth rate by a wide margin over long stretches of time. 3Blanchard and Weil (2001) was originally written around 1990. For other contributions along similar lines, see Gale (1990), Manuelli (1990), Zilcha (1990), Bertocchi (1994) and Barbie, Hagedorn, and Kaul (2004, 2007). 4Limits to arbitrage are crucial also in the literature on irrational bubbles, as emphasized by Shleifer and Vishny (1997). But where we focus on the limits imposed by the nature of the income processes of individuals, that literature has focused more on agency problems in fund management; see Gromb and Vayanos (2010) for a survey. Of course, the two perspectives are complementary. For a recent broader survey of the literature on irrational bubbles, see Scheinkman (2014).

3 nal bubbles and Ponzi-values would disappear. In the model, public debt is always socially preferable to bubbles on private assets, yet eliminating bubbles through high interest rates may not be desirable due to negative effects on capital accumulation. Three natural conditions combine to limit arbitrage in our model:

1. All immortal assets have dividend growth rates below the economy’s growth rate.

2. All mortal assets’ expected unconditional dividend growth rates are below the economy’s growth rate (but could be higher conditional on survival).

3. At any time there are unborn assets, for example future projects that nobody knows about yet, whose rents cannot currently be contracted on.

Condition 1 is well known to be necessary.5 Inasmuch as productivity growth is bi- ased towards physical or human capital rather than land, it is also realistic. Condition 2, or rather the notion of stochastically dying alienable rents, is probably the single most novel component of our model. Despite the fact that in reality (almost) all firms die eventually, we are not aware of any previous work in the bubble literature that considers mortal alienable rents.6 Condition 3, whose relevance to the theory of rational bubbles has previously been discussed by Tirole (1985, p1508), is the second key difference between our model and SW. In SW’s setting, ownership of securities that are currently traded entails ownership of any future securities; it is as if entitlements to all future innovations are effectively owned by existing firms. While Condition 2 is new to the bubble literature, we note that Garleanu,ˆ Kogan, and Panageas (2012) recently invoke both Conditions 2 and 3 in their (bubble-free) analysis of asset price puzzles. Their purpose is to understand the level of risk-premia and the heterogeneity in returns across firms based on the competition from new en- trants – the replacement risk – that the firms face. In our model, on the other hand, the role of replacement risk is to make existing assets sufficiently poor stores of value as to prevent arbitrage over an infinite horizon. To what extent is Condition 3 empirically accurate? As a case in point, consider Facebook, an internet company started by novices in 2003. When it became publicly traded in 2010, the company was worth around 40 billion US dollars. By contrast, in a complete asset market, well diversified investors would already have held financial

5The special case without growth was studied by Scheinkman (1979). The more general condition was established in the context of an OLG model with population growth by Tirole (1985). 6With hindsight, this is perhaps surprising, since depreciation of physical capital plays a central role in both Diamond (1965) and Tirole (1985). Woodford (1990) can be seen as a model of short-lived inalienable rents.

4 claims on Facebook, and any other venture that the founder Mark Zuckerberg may have initiated, long before 2003. Together, Conditions 1-3 generate an economic environment in which dynastic households cannot preserve their wealth forever, and where the crucial contracting friction is caused by overlapping generations of assets (OLGA) rather than overlapping generations of agents (OLG).7 In our model, it is straightforward to show that each of the three conditions are necessary for the existence of bubbles. If we were to simplify the model enough, it would also be possible to precisely characterize a set of sufficient conditions. But we pursue a different route. We ascertain whether sizable bubbles are plausible. Can bubbles be quantitatively important in a state-of-the-art macroeconomic model that does not unduly restrict the set of assets or contracting opportunities? And if so, what is the connection between bubbles and Ponzi-schemes in such a rich model? A major point of our analysis is that the existence of well-functioning stock markets does not preclude rational bubbles. However, we do not claim that rational bubbles are likely to emerge in the stock market itself. The long-run real return to holding stocks has tended to exceed the economy’s growth rate, and this is inconsistent with the existence of a rational bubble on a broad index of stocks. Rather, our analysis should be taken to show why there could be bubbles on lower-yielding assets, such as land, metals, art, and bonds with infinite maturity (consols) – and possibly on certain stocks with a low risk premium – despite the fact that the model’s asset structure is realistically rich.8 The paper proceeds as follows. Section 2 introduces the basic concepts in a partial equilibrium setting. It also pre- empts some common objections to the theory of rational bubbles. Section 3 is a simple general equilibrium model of bubbles that highlights the role of our key assumptions – the mortality of firms and the existence of rents. Section 4 incorporates such a stock market into an otherwise standard general equilibrium model of a production economy with capital, labor, and profits. More

7The conventional wisdom once was that OLG, or more generally the addition of new agents over time, is crucial for the existence of rational bubbles and Ponzi-schemes; see, for example, Blanchard and Fischer (1989, page 227) and O’Connell and Zeldes (1988). However, as shown already by Bewley (1980,1983), there can also be bubbles in models with a constant set of agents, as long as there are enough trading frictions and endowment fluctuations. 8This observation begs the question of whether our model will have anything new to say about the large observed swings in broad-based stock prices. The answer is affirmative. In a follow-up paper we demonstrate how bubble movements involving other assets (for example land and public debt, or on smaller segments of low-yielding stocks) can have great impact on all stock prices due to debt-deflation effects or nominal rigidities in output or labor markets. Technically, then, this is the real economy affecting the broad stock market rather than the broad stock market causing havoc in the real economy – but the shock to the real economy starts with a bubble movement.

5 specifically, our larger model extends the dynastic general equilibrium model of incomplete markets studied by Bewley (1980, 1983), Imrohoro˙ glu˘ (1989), Ayiagari and Gertler (1991), Huggett (1993), and Aiyagari (1994) in two directions. We introduce an incomplete stock market of the kind described above, and we introduce an uninsurable labor income process that is persistent enough to emulate the life-cycle savings motive, in a similar fashion to Castaneda,˜ D´ıaz-Gimenez´ and R´ıos-Rull (2003). The incomplete stock market generates uninsurable entrepreneurial rents. The presence of such rents increases asset demand, whereas the stock market itself increases asset supply compared to the baseline rent–free model. The persistent labor income process, on the other hand, only creates additional demand for assets. Under reasonable parameters, these two additional sources of uninsurable risk suffice to generate a real interest rate below the rate of growth, even after accounting for a large non-fundamental asset price component. That is, the two extensions of the basic model suffice to generate realistically low interest rates without assuming an implausibly incomplete asset market.9 Section 5 studies the model’s quantitative implications as well as implications for fiscal policy. The main finding here is that the model can account for the low (risk-free) interest rates that we observe even if non-fundamental values are large. That is, rational bubbles as well as indefinite roll-over of public debt are plausible. Contrast this finding with Aiyagari and McGrattan (1998) and Floden´ (2001); they introduce public debt in this kind of framework and evaluate the effect of the debt level on social welfare. However, they do so in models where the real interest rate on government bonds exceeds 4 percent, which precludes bubbles and is likely to underestimate the desirability of public debt; in reality, the long-run average real interest rate on US government bonds is about 1 percent. In a sense our approaches suffer from polar op- posite shortcomings: Aiyagari and McGrattan (1998) and Floden´ (2001) overestimate the cost of funds for the public sector, whereas we underestimate the cost of funds for the private sector. This tension will only be resolved once we have a model that offers a credible explanation for the heterogeneous rates of return, a challenge that we leave for future work.10 In our preferred calibration, public debt is highly useful. It facilitates precautionary saving by keeping interest rates higher than they would otherwise be; indeed, the model’s long-run elasticity of the interest rate to the level of public debt is shown to be in the vicinity of empirical estimates, such as Laubach (2009). Also, the public debt

9Krusell, Mukoyama, and Smith (2013) show that aggregate shocks can also generate low risk-free rates in a related calibrated model, but their model has a much more sparse asset structure than ours and is hence more vulnerable to the SW-critique. 10Again, the model of Garleanu,ˆ Kogan, and Panageas (2012) may suggest a promising path.

6 facilitates more insurance. Paying off the public debt would hence entail welfare losses corresponding to several percent of permanent consumption, regardless of whether the agents respond by inflating or deflating private bubbles. Section 6 briefly connects to three other strands of related literature: (i) The literature on bubbles when investment is financially constrained (e.g., Farhi and Tirole, 2012), (ii) the literature on bubbles and self-enforcing debt (especially Hellwig and Lorenzoni, 2009), and (iii) the literature on empirical measurement of rational bubbles (especially Giglio, Maggiori, and Stroebel, forthcoming). Section 7 concludes and indicates some directions for future work.

2 Bubbles and Ponzi-values in Partial Equilibrium

We here present, in a partial equilibrium setting, some basic insights about rational bubbles and Ponzi schemes. Many of the concepts and results are standard, for example from the textbook treatment of Blanchard and Fischer (1989, Chapter 5), but we add some new definitions that will be useful for explaining our subsequent results, in particular the relationship between bubbles and Ponzi-schemes. Let us begin with a caveat. Bubbles and Ponzi schemes are social phenomena. They have value today only because they are widely believed to have value in the future. Therefore, an individual cannot unilaterally decide to start a bubble or maintain a Ponzi scheme. So far, there is no established theory of precisely how and when rational bubbles start. To fix ideas, think about an infinitely lived issuer of a financial asset, inhabiting an economy that grows at a constant rate g and with an exogenous real interest rate r (in subsequent sections, we study general equilibrium models where the crucial point is to endogenize r). The issuer trades with investors (asset holders), and we assume that all parties discount future payments at the same rate. Let pt denote the real market price of the asset at date t. Let dt be the real dividend or interest payment that the asset promises to deliver at date t, and let vt denote the net present value of the asset’s promised dividends. For now, let all of the payments be real; we discuss purely nominal payments below. c Let vt denote the net present value of the real sacrifices that the issuer must incur, in c equilibrium, in order to fulfill the dividend promises; i.e., vt is the issuer’s opportunity cost when funding dividends associated with the current contract. The fact that these sacrifices may differ across equilibria is crucial. c We define the asset’s fundamental value as max {vt, vt } . Accordingly, we define

7 non-fundamental values as follows:

• The asset’s bubble is pt − vt.

c • The asset’s Ponzi-value is pt − vt .

c • The asset’s non-fundamental value is pt − max {vt, vt } .

Suppose first that the financial asset is real debt with finite maturity. The Ponzi- value of the debt contract is the difference between the market price that investors (here, lenders) are willing to pay for it and the net present value of the sacrifices that the issuer (here, borrower) must incur in order to make the contracted payments. We say that the debt is backed if the issuer is always able to repay it (at the market rate of interest) in finite time, i.e., the borrower does not need to rely on indefinite refinancing.11 In an equilibrium where the issuer pays down the debt exactly as specified in the c contract without refinancing any of it, vt = vt, so the debt has zero Ponzi-value. Thus, a fully rational Ponzi scheme requires an indefinite horizon. Either the debt itself must be infinite, or the plan must be to roll it over indefinitely. Could we have a balanced-growth equilibrium in which some debts are rolled over forever? That is one of the central questions that we address in the subsequent sections. (Briefly, the answer is typically affirmative: Real government debt can have positive Ponzi-value.) To begin with, we simply note the immediate implications that follow if the answer is affirmative: If r ≤ g, the issuer’s sacrifices are zero along a balanced growth path, and the aggregate Ponzi-value of an economy equals the market value of those debts that c are expected to be rolled over indefinitely. Then, vt = 0 and the Ponzi-value of a real debt equals its market price pt. When r < g, issuers of debts that are rolled over indefinitely would therefore ben- efit greatly in such an equilibrium. Someone who issues new debt at the rate g can consume a fraction g − r of the accumulated debt every period. Along a balanced growth path, the value of such a debt is wholly non-fundamental if r ≤ g. If r > g some sacrifice is required in order to service the debt. In fact, the discounted value of these sacrifices, at the rate r, exactly equal the value of the debt, which is thus entirely fundamental.

11A stricter criterion would be to require that the debt could be paid down even if all roll-over is denied, in which case the maturity structure becomes crucial. If we additionally were to consider the possibility of strategic default, we would need to deﬁne a concept of credible backing, specifying the willingness (off equilibrium) of the borrower to pay down the debt in ﬁnite time.

8 Suppose next that the ﬁnancial asset is an indeﬁnitely lived stock (share). Specif- ically, suppose the asset issuer owns a productive resource. The resource has a constant survival probability σ, and every period t that it survives it yields a dividend t dt =(1 + g) d0. Finally, let us assume that

σ(1 + g) < 1 + r, (1)

and that the idiosyncratic risk associated with the stock’s survival can be perfectly di- versiﬁed. As will soon be clear, condition (1) is necessary in order for the fundamental value of a stock to be ﬁnite. Analogously, when we come to the general equilibrium model with an endogenous r, this is the consideration that explains why the market must be incomplete (σ < 1) in order for the model to admit r < g in equilibrium.12 Since investors discount future dividend streams at rate r, the fundamental value of the stocks equals the present value of the dividends

σd σ2(1 + g)d v = t+1 + t+1 + ... t 1 + r (1 + r)2 σd = t+1 , (2) 1 + r − σ(1 + g)

where the last equality is a consequence of (1). Rational expectations do not require that the price of a stock equals the fundamental value, however. Rather, in a non-

stochastic rational expectations equilibrium the price pt needs to satisfy the ﬁrst-order difference equation σ(p + d ) p = t+1 t+1 , (3) t 1 + r which, when σ(1 + g) < 1 + r, has the general solution

pt = vt + bt, (4)

with vt satisfying (2) and {bt} satisfying the recursion

σb b = t+1 . t 1 + r

The variable bt is the stock’s rational bubble. In other words, the expected value of next period’s bubble, σbt+1, equals (1 + r) times this period’s bubble, bt. As shown by Blanchard and Watson (1982), there could also be rational expectations equilibria with

12However, our model will be nearer to SW than to Bewley-Ayiagari-Huggett in the sense that we will be considering values of σ that are much closer to 1 than to 0.

9 stochastic bubbles, but we shall not study such equilibria here.13 In other words, we study bubbles that are not expected to burst. While in this model a rational bubble grows at the real rate of interest, the bubble on a surviving stock grows at the rate (1 + r)/σ − 1. Thus, if the bubble and dividend processes are the same across generations of firms, the ratio of the bubble to the fundamental value must be larger on older stocks, as we can only have bubbles when (1 + r)/σ > (1 + g). However, there will be nothing in our model guaranteeing that bubble processes will be the same across generations of firms, or even within them. For example, the model would allow there to be a bubble on new “high-tech” stocks even if there is no bubble on other new firms, and even within this class of relatively young assets, the relationship between age and bubble size could be weak if g is close to r/σ. As indicated above, we think that in modern economies conditions rarely admit a bubble on the broad stock index. Average returns are well above the growth rate. However, equilibrium returns could be small enough on stocks that provide a hedge against risks that could hit average stocks hard. Notice that everything we have said about stocks so far is also applicable to real assets such as land. To the extent that land lives forever, simply set σ = 1. The key condition for bubbles to exist on land, equation (1), then becomes gl < r. That is, the growth rate of the real dividend from land, gl, must be smaller than the economy’s real rate of interest, r. We can thus have rational bubbles in the economy as long as 14 gl < r < g, where g is the economy’s average rate of growth. To the extent that productivity grows faster for other input factors (capital, labor, ideas) than for land, this condition is reasonable. Clearly, we could in principle have a bubble also on land that yields zero fundamental value. By extension, the there could be rational bubbles on all kinds of unproductive assets. Rare stamps, paintings, and other expensive collectors’ items are relevant examples. As noted by Tirole (1985), if these unproductive assets are infinitely lived and r < g, the aggregate value of such unproductive assets can only remain a significant fraction of the economy’s overall asset value if there is steady creation of new assets. Antiquities and art are two categories of assets that naturally fulfill the criterion of such a steady inflow.

13If agents are risk-neutral or bubbles can be perfectly diversiﬁed, the analysis is quite straightforward, with many expressions simply being replaced by their expected value. 14For a while, many economists thought that the existence of an inﬁnitely-lived productive asset would make impossible the emergence of rational bubbles, but with Tirole (1985) it became clear (as the above analysis shows) that the existence of productive land only precludes rational bubbles in an economy without growth.

10 Consider finally nominal public debt. In practice, nominal debt is always paid for by the government’s own money printing. This is hence a promise that involves no explicit claim to real resources. As vt = 0, it is entirely a bubble, regardless of the nominal interest on the debt and of the duration of any specific security. However, unlike other bubbles, it can be sustained even if r > g. Along a balanced growth path, the government can use taxation in order to fund the interest payments. But since taxes are costly, there is no longer any Ponzi-value. Thus, the non-fundamental value is now zero, just as in the case of real debt. A major attraction of rational bubble theories is that they are consistent with several asset pricing facts that contradict standard “fundamentalist” models. Importantly, rational bubble theories account for such anomalies without invoking either irrationality or unrealistic constraints on short-selling. For example, the model of rational bubbles is consistent with the observation that assets with identical dividend streams are sometimes traded at markedly different prices (Rosenthal and Young, 1990; Mitchell, Pulvino, and Stafford, 2002). It is a widespread misunderstanding among academic economists that different prices for identical assets cannot occur in a rational-expectations equilibrium whenever short-selling of the more expensive asset is allowed. This is incorrect, because short-selling requires collateral, both in our theory and in practice.15 Once the short- seller nears the borrowing constraint, which always eventually happens in a model with independent uninsurable shocks such as ours, the position must thus be closed. If the price-ratio between the two assets has not changed by then – and it will not have in the equilibria that we study – there is no arbitrage profit, even in the extreme case of financial markets with zero haircuts.16 Moreover, since only the aggregate bubble is pinned down by the model’s deep parameters, there is ample scope for price-variation that is orthogonal to changes in dividends; this is the bubble-substitution theory due to Tirole (1985). Indeed, we think that the bursting of rational bubbles is a natural candidate for some of the “rare disasters” in asset markets (Rietz, 1988; Barro, 2006), but we refrain from analyzing such episodes in this paper.

15For example, the Federal Reserve Board (under Regulation T) requires all short sale accounts to have 150 percent of the value of the short sale at the time the sale is initiated, and while subsequent maintenance margins can be lower, collateral requirements typically exceed 125 percent of the shorted asset’s market value. 16In deterministic economies, on the other hand, bubbles can often be ruled out, as shown by Tirole (1982). However, this result also comes with qualiﬁcations; see Kocherlakota (1992).

11 3 An Incomplete Stock Market

To clarify what we mean by an incomplete stock market, and explain why such a stock market can be consistent with large non-fundamental asset values in general equilibrium, we first introduce the central features of our analysis within a simple laissez-faire endowment economy. Since there is no investment, such an economy trivially satisfies the efficiency criterion of Abel et al (1989). Time is discrete, and the horizon is infinite. Period t = 0 refers to the current period. Periods t = −∞, ..., −1 comprise the history and determine the “initial conditions” that characterize period 0. Periods t = 1, ..., ∞ comprise the future. There is a continuum of infinitely lived agents distributed along the unit interval.17 Preferences: Agents consume a homogeneous good and have identical preferences. Their utility function is ∞ t U = E ∑ β u (ct) , (5) t=0 where ct is consumption in period t, and β ∈ (0, 1) is the subjective discount factor. We assume that the felicity function u takes the CRRA form

c1−µ u(c)= . (6) 1 − µ

Technology: There are two kinds of productive assets. First, there is a continuum of length A of productive and tradable Lucas trees. In each period that a tree is alive, it yields an amount d ≥ 0 of non-storable fruit. Survival is an i.i.d. process; each tree survives to the next period with probability σ < 1. Thus, a measure (1 − σ)A trees die each period. Likewise, a measure (1 − σ)A trees are born each period. Of the trees that emerged v years ago, there thus remain (1 − σ)σv A. Each new tree is paired with a random agent, who initially has full ownership of this tree. For convenience, we assume that an agent can absorb at most one new tree in any period. Thus, in each period a fraction 1 − σ of the agents receive a new tree. Let εt be an indicator variable, taking the value 1 if an agent gets one of the new trees in period t and 0 otherwise. t t Let ε = {ε0, ..., εt} denote the partial sequence from period 0 up to period t and let E , denote the corresponding set of all possible such sequences. Define probability mea- t t t t sures γ (x0, ·) : E → [0, 1] , t = 0, 1, ... where, for example, γ (x0, ε ) is the probability t of history ε given an agent’s initial state x0 specified below. Second, there is a continuum of length 1 of infinitely lived non-tradable trees each yielding y units of fruit per period. Each agent owns one non-tradable tree; to dis-

17As usual, each inﬁnitely lived agent could be seen as representing an altruistically linked dynasty.

12 tinguish them from the tradable trees, we refer to them as bushes from now on. The role of the bushes is to keep consumption bounded strictly above zero and utilities bounded above negative infinity, which is convenient for computational purposes. (In reality, we do not think that there exist infinitely lived assets whose dividend pas- sively grows at the economy’s overall rate of productivity; in the next section, the role of these bushes will therefore be played by a social insurance system.) There also exists a continuum of length S of an unproductive asset that has no intrinsic value but which is perfectly durable and can be traded. We call this asset stamps. Trade: At any time, agents can trade fruit, stamps and stocks in existing trees in a frictionless market. In principle, they are also able to borrow against collateral (shares in trees), but not against the income from bushes. Borrowing constraints are a central feature of the Bewley-Aiyagari-Huggett framework. As will become clear, our main results do not hinge on zero borrowing; they hold even if agents are allowed to have realistically negative asset holdings. Moreover, it is straightforward to justify realistic borrowing constraints from first principles by assuming that agents can hide incomes from the bush (move to another jurisdiction) or that debts are not inherited within a dynasty comprising finitely lived but altruistically linked representatives. Finally, we assume that claims on trees that have not yet emerged cannot be traded. While agents could in principle write contracts contingent on future ownership of trees that have not yet emerged, our central assumption is that such contracts will not be worth writing. We find the assumption realistic, and the ensuing analysis does not hinge on its precise justification, but let us nonetheless briefly indicate one reason why it is difficult to contract on non-existing assets: Suppose that the paired agent privately observes the tree’s emergence one period before other agents do so. The agent can choose whether to keep the emerging tree or covertly display and sell it to some other agents – potential buyers – before it becomes publicly visible. Suppose the seller has sold shares (stocks) in own emerging trees, but some other agent has not. That agent can then buy the tree, pretend it emerged with him, and cash in the full market value of the tree in the next period. By contrast, the seller by waiting until the next period can only cash in the value of any retained stocks. Thus, a coalition of agents will always find it profitable to deviate from any plan involving trade of claims on future trees. More precisely, any outcome in which a positive measure of agents share risk in a market for futures would fail to be coalition-proof, in the sense defined for this kind of private-information environment by Lacker and Weinberg (1993). Prices: As we seek to endogenize the value of financial assets, the consumption

13 good, fruit, is our numeraire. That is, the prices of other assets are expressed in terms of fruit. We assume that all units of the unproductive asset are indistinguishable. There are no labels or other extraneous features that may be used to distinguish them. Thus, they must be priced the same. The price of one unit of the unproductive asset at time s t is denoted pt. While the productive assets are also intrinsically identical, in the sense that each tree yields the same expected future return, their age differs. We thus allow the price

to differ across generations of trees, letting pj,t denote the price of a tree of generation j ≤ t at time t.18 Allowing assets from different generations to be priced differently facilitates the study of price bubbles on productive assets. We do not consider different prices for assets within the same generation, but there is nothing in the model that prevents such equilibria.

Agents’ wealth: Let st denote the quantity of unproductive asset that an agent pos-

sesses at the end of period t − 1. Let aj,t denote the quantity of generation j assets that an agent possesses at the end of period t − 1, and let

= t−1 at aj,t j=−∞

denote an agent’s holdings of all productive assets born in previous periods. Let x0 = ∞ (a0, s0, ε0) ∈ X denote the initial state of an agent, where X = R × R × {0, 1}. Let X = R∞ ×R× {0, 1} where R denote the Borel sets that are subsets of R. (This set includes the set of all possible relevant states for the economy.) Observe that we admit negative asset holdings. Thus, short-selling is allowed, as long as the agent does not violate an overall borrowing constraint to be speciﬁed below.

Behavior: In period 0, given its initial state x0, each agent chooses consumption t t and savings for each possible sequence ε . Let φt : E → Xas, t = 0, 1, ... describe the ∞ t savings plan, where Xas = R × R and φa,j,t(ε ; x0) denotes the value for aj,t+1 that is chosen in period t if the history up to t is εt, conditional on the agent’s initial state t being x0. Similarly φs,t(ε ; x0) denotes the value for st+1. The savings plan must satisfy a no-borrowing constraint

t s ∑ pj,taj,t+1 + pt st+1 ≥ 0, (7) j=−∞

18It is not necessary that agents keep track of the assets’ age, t − j. In order to sustain different prices for assets belonging to different generations, it sufﬁces that agents recall the assets’ price in the previous period.

14 t t t where aj,t+1 = φa,j,t(ε ; x0) and st+1 = φs,t(ε ; x0). Let ct : E → R+ describe the associated plan for consumption. At the beginning of period t, a fraction σ of the trees alive at t − 1 will have died; the remaining will yield a dividend d. An agent’s budget constraint is therefore given by

t−1 t t s s ct(ε ; x0)= y + pt st + ∑ (pj,t + d)σaj,t + (pt,t + d) εt − ∑ pj,taj,t+1 − pt st+1. (8) j=−∞ j=−∞

The agent’s problem is thus to maximize expected discounted lifetime utility

∞ t t t t ∑ ∑ β u ct ε ; x0 γ (x0, ε ) (9) t=0 εt∈E t

t t t through a set of choices ct(ε ; x0) and φt(ε ; x0) for all t, subject to (7)-(8) and ct(ε ; x0) ∈ ∞ t t−1 s R+ and φt(ε ; x0) ∈ Xas and taking as given the sequences of prices pj,t =−∞ , pt n j ot=0 and the initial state x0. Markets and distribution: The distribution of agents over the initial state is described by a measure κ : X → [0, 1]. By integrating over κ, aggregate variables can be com- puted. Market clearing in ﬁnancial markets implies that

S = s0dκ, (10) ZX

−j σ (1 − σ)A = aj,0dκ, ∀ j = −∞, ..., −1. (11) ZX and for all t ≥ 0 that t t t S = ∑ φs,t(ε ; x )γ (x , ε )dκ, (12) Z 0 0 X εt∈E t

σt−j(1 − σ)A = ∑ φ (εt; x )γt(x , εt)dκ, ∀ j = −∞, ..., t. (13) Z a,j,t 0 0 X εt∈E t Similarly, market clearing in the market for fruit implies that

t t t d + y = ∑ ct(ε ; x )γ (x , ε )dκ, ∀ t ≥ 0. Z 0 0 X εt∈E t

∞ t−1 s Equilibrium: An equilibrium comprises sequences of prices pj,t , pt j=−∞ t=0 t t ∞ n t t o and sequences of decisions ct(ε ; x0), φt(ε ; x0) t=0 for all x0 ∈ X and ε ∈E , together t ∞ t with probability measures γ (x0, z) t=0 for all x0 ∈ X and z ∈E , and a measure κ(x) 15 for all x ∈X describing the initial distribution such that

1. the decision rules solve the agents’ problem given prices and the initial state x0;

2. all markets clear, and

t t t t−1 t−1 t−1 t t−1 3. the measure γ (x0, ε ) satisﬁes (i)γ (x0, ε , 0 )= σγ (x0, ε ), (ii) γ (x0, ε , 1 )= t−1 t−1 0 0 (1 − σ)γ (x0, ε ) for all x0 ∈ X, t ≥ 0, and (iii) γ (x0, z) = 1 if ε0 ∈ z ∈ E and 0 otherwise.

3.1 Analysis

We will be mostly concerned with stationary equilibria, where returns to saving and the aggregate distribution of wealth are constant, both with respect to agents and asset classes. Note that individual wealth may be highly variable even if the aggregate distribution is constant. Indeed, the agents will optimally engage in precautionary saving precisely because they are concerned about the negative consumption consequences of long strings of poor luck. a Let Rj,t = σ(pj,t+1 + d)/pj,t denote the return to holding a tree of generation j s s s and Rt = pt+1/pt denote the return to holding stamps. Since there is no aggregate uncertainty and all existing trees give fruit in any equilibrium, no-arbitrage implies a a s a that Rj,t = Rt . Moreover, if stamps are valued no-arbitrage implies that Rt = Rt . Stationary equilibria hence come in three types.

No-bubble equilibrium For all σ ∈ (0, 1] there exist a stationary equilibrium without s any price bubbles on stamps or trees, where p = 0, φs(·; x0) = 0 for all x0 ∈ X, > a a and pj = p 0 such that Rj = R ≥ σ for all j.

Concentrated bubble equilibrium For all σ ∈ (0, σ¯ ] there also exist a stationary equi- < σd librium, in which stamps are the only bubble, where σ¯ 1, pj = 1−σ for all j and s > a s p 0 such that Rj = R = 1 for all j.

Dispersed bubble equilibrium For all σ ∈ (0, σ¯ ] there also exist a continuum of sta- s tionary equilibria with price bubbles on trees, where p = 0, φs(·; x0) = 0 for all a a a x0 ∈ X, and pj,t+1 = (R /σ) pj,t − d > 0 where R < R < 1 for all j and t, and pt,t = p.

We use the terms concentrated and dispersed to denote if the bubble is located on a single asset or on multiple assets. In the concentrated bubble equilibrium, the bubble is

16 located on stamps and the value of the aggregate bubble is simply psS. In the dispersed bubble equilibrium, the bubble is located on trees of different generations. The bubble σd σd on a new tree is p − Ra−σ where Ra−σ is the fundamental value of a tree. Over time the bubble on an individual tree grows at rate (Ra/σ) − 1 > 0. But since trees die at rate (1 − σ) the total bubble on a generation of trees falls at rate Ra − 1 < 0. σd a The aggregate bubble is thus (1 − σ)A p − Ra−σ / (1 − R ). Moreover, the oldest existing tree carries the largest bubble in the economy, but the youngest generation of trees carries the largest share of the aggregate bubble on trees. (The latter property is a consequence of the class of equilibria that we focus on and not a general property of the model. For example, the model admits equilibria in which bubbles stochastically move from some classes of trees to others; the observation of a large crash on relatively young stocks does not refute the theory of rational bubbles.) The absence of a complete set of Arrow-Debreu contingent claims markets is crucial for the existence of bubble equilibria. To see this, note that if there was a market in period t where agents could buy and sell shares of trees that do not exist in t but may exist in t + 1, then a no-arbitrage condition would imply identical returns to a diversiﬁed portfolio of existing trees and to one of non-existing trees. The uncertainty surrounding new trees would then be insured away; there would be no consumption uncertainty, the equilibrium distribution of wealth would be degenerate, and the rate

of return would be given by 1/β. The equilibrium price of trees would then be pj,t = σd 1/β−σ for all j, t. In order to illustrate the conditions under which there will be bubble equilibria, and to study their welfare properties, let us now impose additional structure.

3.2 Parametrization

As we will be using some of the parameter values later, we discuss these choices now, even if the simple model itself is too rudimentary to be taken seriously for quantitative purposes. Let a period be one year, and set the utility discount factor β to 0.97.19 Kimball, Sahm, and Shapiro (2009) provide some of the most recent estimates of the coefficient of relative risk aversion µ. Using survey results in the PSID they find that the average µ is 4.19 (see their Table 1). Since our main results are generally strength- ened as µ increases, and we prefer to be conservative, we set µ = 3. For convenience, we normalize S = A = 1. 19It is well known that the discount factor has been difficult to pin down exactly in empirical studies (Frederick, Loewenstein, and O’Donoghue, 2002). For this reason, we will calibrate it to match the wealth/income ratio when pursuing the quantitative investigation in Section 4.

17 The yield from trees is normalized to d = 1. Recall that bushes can not be traded. The yield from a bush can hence be thought of as the level of safety nets in the economy. We set it to y = 0.1 units of fruit each period. With average consumption equal to d + y, the poorest agent in the economy, one with zero assets, can thus consume approximately 9 percent of average consumption. As a comparison, having an income of 9 percent of average income in the US would place an individual in approximately the bottom 5th percentile (see D´ıaz–Gimenez,´ Glover and R´ıos-Rull, 2011).20 The final parameter, σ, governs the rate of creation and destruction in the economy. Jovanovic and Rosseau (2005) find, using US data between 1885 and 2003, that the value of newly listed firms has on average been just above 3.1 percent of total stock market value, with highs of above 10 percent in periods of rapid technological change such as the electrification era and the IT era. In the period in between these eras, the value of newly listed firms gradually increases; between 1930-1979 the average value was 1.4 percent of total stock market value and between 1980-2003 it was 3.6 percent. As the criteria for being listed on the stock exchanges are quite stringent, these numbers can be thought of as a loose lower bound for new firm creation.21 To illustrate the mechanisms at work, we shall display results for all values for σ in the range [0.85, 1]. We compute the solution to the agent’s problem using Carroll’s (2006) endogenous grid method. Since the returns to stamps and a perfectly diversified portfolio of trees are equal in any equilibrium with valued stamps, agents are indifferent between holding trees and stamps in such equilibria. For simplicity we thus assume that all agents hold the same portfolio shares. We then compute the stationary distribution by ap- proximating the invariant density function.

20Also, as of 2010, the maximum monthly allotment of food stamps for one person in the US is $200 per month. According to the Bureau of Labor Statistics, average annual expenditure for single individual households was $30,613 in 2011, which implies that food stamps provide a safety net of slightly less than 8 percent of average consumption. 21According to Caves (1997), who considers a broad sample of firms in eight different countries during the 1970s and early 1980s (using data assembled by John Cable and Joachim Schwalbach), average annual entry rates are 7.7 percent in the US and as high as 13 percent in Belgium. Average exit rates are very similar to entry rates. Using more recent data covering 24 countries, Bartelsman et al. (2004) find that gross turnover (exit plus entry rates) are between 20-25 percent across all firms. Among large firms, defined as those with at least 20 employees, exit and entry rates are around five percent. In the time series dimension, they find that between 15-20 percent of firms in industrial countries fail during the first two years, that across all countries 65 percent of firms survive the first four years while only 30-50 percent of all firms survive beyond seven years. This points to an average yearly rate of destruction/creation of 8 percent.

18 3.3 Equilibria

Figure 1 summarizes the main results. It displays the equilibrium prices for yearly survival rates σ in the range [0.85, 1] in the three different classes of equilibria; (i) without bubbles (ii) with bubbles concentrated on one asset (stamps) and (iii) with bubbles on multiple assets (trees of different generations). As a benchmark, the ﬁgure also displays equilibrium prices under complete markets.

A 45

40 Price of trees (w/o bubbles)

30 Price of trees (conc. bubble) Price of new trees 25 (dispersed bubbles)

price B

20 Price of trees w. complete markets

10 Price of stamps

5 (concentrated bubble)

0 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 Survival rate

Figure 1: Asset prices as functions of the survival rate σ

With complete markets, the price of trees falls as their survival rate falls, since the discounted stream of dividends falls. With incomplete markets but no bubbles, the price of trees ﬁrst increases and then falls as the survival rate is reduced. The non-monotonicity reﬂects two competing effects. On the one hand, as the survival rate falls trees become less valuable since the expected dividends are smaller. On the other hand, without perfect insurance markets the demand for precautionary saving increases. For high survival probabilities the latter effect dominates. But as the survival rate decreases, the chance of receiving a new tree increases. The precautionary motive is reduced, and the price of trees falls. The strength of the precautionary motive depends on the relative size of the bush and and the dividends from trees; the larger the bush, the smaller is the precautionary motive. For a large enough bush, the non-monotonicity disappears.

19 A concentrated bubble equilibrium exists for all survival rates σ < 0.979945. With a bubble on stamps, the price of trees monotonically falls as the survival rate falls, just as under complete markets. This is natural; in both cases we are discounting a diminished stream of dividends at a constant interest rate (the interest rate r = Ra − 1 is always zero in the concentrated bubble equilibrium, and 1/β − 1 = 0.03 in the complete markets economy). For all survival rates σ < 0.979945 there also exist dispersed bubble equilibria. The shaded area in Figure 1 shows the price of new trees in these equilibria. For example, consider all dispersed bubble equilibria on the vertical line between points A (the no bubble equilibrium) and B (the concentrated bubble equilibrium) in the ﬁgure. In A the price of trees is high, implying a low real interest and thus a high fundamental σd value of a new tree, Ra−σ . The low real interest implies low aggregate saving, and in A the real interest rate is so low that it rules out bubbles. A downward movement from A to B is associated with a lower price on new trees but a higher real interest rate. This implies that the fundamental value of a new tree monotonically falls between A and B. The bubble value of a new tree however changes non-monotonically. In A there are no bubbles, and in B the bubble is concentrated on stamps. Hence, the bubble on new trees ﬁrst increases and then decreases as we move from A to B. But even if the bubble on each new tree is smaller in the equilibria close to B, the real interest rate is higher, and the bubble on a generation of trees thus falls at a slower rate. Therefore, the aggregate bubble increases monotonically as we move from A to B and is largest in B where stamps are the only bubble. Finally, in terms of ex-ante average welfare these equilibria can easily be ranked; the larger the aggregate bubble the higher is average welfare. The larger are the bubbles, the higher is the return to saving, and the smaller is the consumption inequality.

4 A Medium-Sized Macro Model

To quantitatively investigate how our analysis carries over into a more realistic calibrated model, we now embed our incomplete stock market into a standard macro model with capital and labor. More speciﬁcally, our larger model extends the standard dynastic general equilibrium model of incomplete markets by (i) introducing an incomplete stock market of the kind described above, and by (ii) introducing an uninsurable labor income process that is persistent enough to emulate the life-cycle savings motive, in a similar fashion to Castaneda,˜ D´ıaz-Gimenez´ and R´ıos-Rull (2003). We model proﬁts (rents) in the conventional way of having a monopolistically com-

20 petitive sector where all firms earn the same mark-up in equilibrium. We also introduce a government that issues nominal debt, paying a nominal rate of interest, and that levies taxes and transfers. For the purposes of the analysis in the present paper, we might as well have assumed that public debt were real. However, since we aim to exploit the model to analyze monetary phenomena in future sequels, we introduce nominal bonds already now. Time is discrete, the horizon is infinite and period t = 0 refers to the current period. There is a continuum of infinitely lived agents distributed along the unit interval. Preferences: As before, we assume that agents consume a homogeneous final good and have identical preferences as described by (5) and (6). Inalienable endowments: (i) Labor. Each period, an agent is endowed with one unit of labor. Labor has no opportunity cost, and is thus supplied inelastically. The vari- w w w able εi,t indicates whether a agent i is productive (εi,t = 1) or not (εi,t = 0). We think of the unproductive state as retirement and the productive state as working-age. Pro- e ductivity follows a first order Markov process. (ii) Ideas. The variable εi,t denotes e whether an agent i is endowed with an entrepreneurial idea in period t (εi,t = 1) or e < not (εi,t = 0). An idea survives to the next period with probability σ 1. Obsolete ideas are replaced with new ideas. Each new idea is paired with an agent who then becomes an entrepreneur. Let A denote the continuum of ideas. Technology: (i) Final goods. The production of final goods is conducted by competitive producers, and requires neither ideas nor labor. Instead, it requires a continuum of intermediate goods. The final goods production function is

A 1/η η Yt = (yn,t) dn , Z0 where yn,t is an intermediate input, and where 0 < η ≤ 1. (ii) Intermediate goods. The intermediate goods sector consist of a continuum A of monopolies. Each monopoly n requires a unique idea that allows it to produce a speciﬁc intermediate good by combining physical capital k and labor h. The capital is in the form of ﬁnal goods. The intermediate goods production technology is

α 1−α yn,t = (kn,t) (λthn,t) ,

t where labor productivity λt = (1 + g) grows at an exogenous rate g. If an idea becomes obsolete, the ﬁrm ceases to exist. All intermediate goods producers are initially private companies, and the agent owning the idea bears all the risk. Each period, each entrepreneur is enabled with

21 probability σo to (costlessly) issue shares (stocks) in their ﬁrm and transform it into a o public company. Let εi,t denote whether agent i received an opportunity to make an o o IPO (εi,t = 1) or not (εi,t = 0). Let A1 denote the continuum of private companies and A2 the continuum of public companies, where A = A1 + A2. Assets and asset holdings: There are four kinds of tradable assets: (i) Capital corre- 22 k sponds to the renting out of ﬁnal goods to be used as an input in production. Let rt be the market rate of return to capital at t, and let δ be the capital depreciation rate. (ii)

Stocks are financial claims on the profit dn of an intermediate goods producer n, to be described below. (iii) Government bonds constitute a promise by the authorities of a b fixed nominal repayment 1 + rt every period but the last, when the nominal principal b is also repaid. That is, the interest on a bond, rt is expressed in terms of bonds rather than final goods. To ease notation, we assume that the bonds have infinite duration (they are consols), and hence are repaid whenever the government desires to do so.23

Thus, let Bt denote the amount of government bonds issued at t − 1, where Bt ≥ 0. (iv) Stamps are unproductive assets that have no intrinsic value but which are perfectly durable and can be traded. One interpretation is that stamps correspond to a subset of the land in the economy. Let St denote the volume of stamps at the end of s period t − 1, where St ≥ 0. In each period t an amount gt St new stamps appear so s s 24 that St+1 = (1 + gt )St; the value of gt will be determined in equilibrium. Each new s stamp is randomly paired with an agent. The variable εi,t denotes whether an agent i s s is endowed with a new stamp (εi,t = 1) or not (εi,t = 0). Let ki,t, bi,t and si,t denote agent i’s holding of physical capital, bonds and stamps respectively, and let ai,n,t denote the agent’s holding of stocks in firm n. Trade: There are frictionless markets for final goods, intermediate goods, labor, and assets. We express all prices and payments (except the interest on government bonds) in terms of final goods. Specifically, the price of capital is always 1 (since the final y good is our numeraire), wt is the wage at t for a productive worker, pn,t is the price of b s intermediate good n, pt is the price of government bonds, and pt is the price of stamps. For simplicity, we shall assume that investors do not keep track of companies’ birth. At most, they remember the previous period’s market price. Thus, to the extent that companies’ price depend on the company’s history, the dependence only involves

price movements since the period they where listed. Henceforth, let pj,t denote the

22We might alternatively have allowed ﬁrms to own capital and issue the corresponding amount of debt, backed by this capital. 23While we choose consols only for simplicity, and because duration is not important for the issues under study in this paper, consols have also been used in practice. The UK government issued its ﬁrst consols – or undated gilts – in 1751, its last in 1946, and it redeemed all outstanding consols by July 2015; see http://www.dmo.gov.uk/index.aspx?page=gilts/about gilts. 24Under the land interpretation, bubbles thus emerge randomly on new areas of land.

22 price in period t of a public company that was listed in period j. b Government: The government sets the nominal interest rate rt and the growth rate b b gt of the stock of nominal government debt Bt, where gt = Bt+1/Bt − 1. The revenue from changing the stock is used to ﬁnance lump-sum transfers (taxes) to all agents; b b b τt = pt (Bt+1 − (1 + rt )Bt). The government also imposes a proportional labor income tax, τ, to fund government consumption and a proportional pension tax on labor in- p come, τ , to pay for lump-sum pensions θt to retirees. Within-period timing: (i) Production takes place and income is distributed. (ii) A

fraction σo of the hitherto private entrepreneurs get the opportunity to take their firm public. (iii) New stamps appear. (iv) Agents consume and save. (v) New ideas are realized. (vi) A fraction 1 − σ of all ideas become obsolete and the associated intermediate goods firms are destroyed. Thus, the ratio of public to private firms is

A2/A1 = σσo/(1 − σ) and (1/σ − 1) A new ideas are realized each period. The total supply of stock at the end of a period is A2/σ. An agent’s state and expectations: From now on, let us suppress agent indices i. More- over, let us anticipate that agents (since they are risk-averse) will not want to bear unnecessary idiosyncratic risk. Since markets are frictionless, it is optimal to hold per-

fectly diversified portfolios of tradable stocks. Let aj,t denote an agent’s holdings at t of a diversified portfolio of public companies that became first listed in period j, and = t−1 let at aj,t j=−∞ be the agent’s holding of the market portfolio of all tradable stocks. To ease notation, and without significant loss of insight, we henceforth assume that

agents hold the market portfolio of publicly traded assets. Let pt denote the price of the market portfolio. w e o s Let εt = (εt , εt, εt, εt). The current state of an agent can then be expressed as x0 = 4 4 4 4 (a0, b0, k0,s0,ε0) ∈ X, where X = R × {0, 1} . Let X = R × {0, 1} . We assume that the probability of receiving a new stamp is independent of productivity. These events are also independent of entrepreneurship and IPO opportunities. Moreover, for simplicity, we assume that (i) an agent can at most have one currently relevant idea, (ii) an entrepreneur who receives an opportunity to make an IPO can not receive a new idea in the same period, (iii) a newborn ﬁrm can not enter the stock market, and (iv) an agent can at most receive one new stamp per period. Let Γ denote the Markov transition matrix that describes how εt evolves over time.

4.1 Analysis

Production: Since markets are frictionless, and there are no irreversibilities, ﬁrms will solve their static proﬁt maximization problem each period.

23 Final goods producers are competitive and solve the problem

A y max Yt − pn,tyn,tdn. yn,t Z0

From the ﬁrst-order condition, we deﬁne the inverse demand

η−1 y yn,t pn,t(yn,t) ≡ . (14) Yt

Each intermediate goods producer n maximizes proﬁts

y k dn,t = pn,t(yn,t)yn,t − wthn,t − rt kn,t

k by renting capital kn,t at price rt and hiring labor hn,t at price wt, subject to (14). Since each ﬁrm solves the same problem, and that problem has a unique solution, 1/η y y 1/η−1 we can write yn,t = yt. Thus in any equilibrium, Yt = A yt and pn,t = pt = A . From the ﬁrst-order conditions,

k 1/η−1 α (1−α) −1 rt = ηαA kn,t (λthn,t) kn,t , (15) 1/η−1 α (1−α) −1 wt = η(1 − α)A kn,t (λthn,t) hn,t , (16) which implies that proﬁts are given by

1/η−1 α (1−α) dt = A kn,t (λthn,t) (1 − η). (17)

Aggregate labor demand is Ht = Ahn,t, and aggregate demand for physical capital is

Kt+1 = Akn,t+1. Consumption: In the current period, conditional on the current state x0, each agent t t 4 plans consumption and savings for each possible future sequence ε . Let φt : E → R , t t = 0, 1, . . . , describe the savings plan, where φa,t(ε ; x0) denotes the value for at+1 that t t is chosen in period t if the history up to t is ε . Similarly φb,t(ε ; x0) denotes the value t t for bt+1, φk,t(ε ; x0) denotes the value for kt+1 and φs,t(ε ; x0) denotes the value for st+1. The savings plan must satisfy a no-borrowing constraint

b s ptat+1 + pt bt+1 + kt+1 + pt st+1 ≥ 0. (18)

t Let ct : E → R+ describe the associated plan for consumption. At the end of each period, a fraction σ of the public ﬁrms dies. An agent that saved

at in period t − 1 thus has σat at the beginning of t. Labor income is yt = (1 − τ −

24 p p p τ ) wtεt +(1 − εt )θt . An agent’s budget constraint is therefore t b ct(ε ; x0) = yt + τt + (rt + 1 − δ) kt − kt+1 e o +(pt + dt)σat − ptat+1 + dtεt + ptεt b b s s +pt (1 + rt )bt − bt+1 + pt (st + εt − st+1) . (19) t t ∞ The agent’s problem is thus to choose a feasible plan ct(ε ; x0), φt(ε ; x0) t=0 to maximize expected discounted lifetime utility

∞ t t t t ∑ ∑ β u ct ε ; x0 γ (x0, ε ) (20) t=0 εt∈E t subject to (18)-(19), the transition matrix Γ, anticipated sequences of prices, dividends, taxes and transfers, and the initial state x0. This economy features exogenous growth of productivity and endogenous growth in stamps. Moreover, government policy will be determining a growing supply of bonds. To solve the model, it is therefore convenient to normalize variables by their respective growth rate. A tilde denotes that the variable is de-trended. Most variables are de-trended by productivity, for example c˜ ≡ ct . The exceptions are the following; t λt b s pb ≡ pt Bt , b ≡ bt , ps ≡ pt St , and s ≡ st . Note that the de-trending implies that we t λt t Bt t λt t St normalize aggregate supply of bonds and stamps to one in all periods; B = S = 1 for e e e e t t all t. e e Markets: Market clearing in ﬁnancial markets is given by

a˜0dκ = A2, (21) ZX

b0dκ = 1, (22) ZX e k˜0dκ = K˜0, (23) ZX

s0dκ = 1, (24) ZX and for all t ≥ 0 by e t t t ∑ φa,t(ε ; x )γ (x , ε )dκ = A /σ, (25) Z 0 0 2 X εt∈E t e ∑ φ (εt; x )γt(x , εt)dκ = 1, (26) Z b,t 0 0 X εt∈E t e

25 t t t ∑ φ (ε ; x )γ (x , ε )dκ = K˜ + , (27) Z k,t 0 0 t 1 X εt∈E t e t t t ∑ φs,t(ε ; x )γ (x , ε )dκ = 1. (28) Z 0 0 X εt∈E t e Similarly, goods market clearing implies that

t t t Y˜t = ∑ ct(ε ; x )γ (x , ε )dκ + G˜t +(1 + g)K˜ + − (1 − δ)K˜ t. Z 0 0 t 1 X εt∈E t e t t Finally, labor market clearing implies that H = ∑ t t x γ (x , ε )dκ. X ε ∈E ,εt =1 0 Government: To facilitate accounting, and withoutR loss of generality of our results, we depict the government as running three separate budgets. Government expenditure is ﬁnanced through a labor income tax

G˜t = τw˜ t H, (29)

public pensions is ﬁnanced through a pension-tax

p t t p ∑ (1 − τ − τ )θ˜tγ (x0, ε )dκ = τ w˜ t H, (30) ZX t t x ε ∈E ,εt =0 and the lump-sum transfer is financed through the growth of (nominal) public debt net of interest payments b b b b τ˜t = gt − rt pt . (31) ∞ e b s y k Equilibrium: An equilibrium comprises sequences of prices p˜t, pt , pt, pt , rt , w˜ t t=0 , ∞ ∞ ∞ dividends d˜ , government policies rb, gb, τ˜b, τp, θ˜ , τ, G˜ , capital stocks K˜ , t t=0 t t t t t t=0 e e t t=0 s t t ∞ growth rates of stamps {gt } , and decisions ct(ε ; x0), φt(ε ; x0) t=0 for all x0 ∈ X and ∞ for all εt ∈ E t, together with probability measures γt(x , j) for all x ∈ x and for e e 0 t=0 0 all j ∈E t, and a measure κ(x) for all x ∈X describing the initial distribution, such that (i) the decision rules solve the agents’ problem given prices, dividends, policies and y 1/η−1 the initial state x0, (ii) factor prices are given by (15)-(16), and pt = A , (iii) dividends are given by (17), (iv) all markets clear, (v) the government budget constraints t t (29)-(31) are satisfied, and (vi) the measure γ (x0, ε ) is consistent with the transition matrix Γ. Solution: Like the simple model, the full model admits no-bubble equilibria, concentrated bubble equilibria and dispersed bubble equilibria. For simplicity, we restrict our discussion to stationary equilibria. These can be characterized via the no-arbitrage ˜ a ˜ ˜ k k conditions. Let the variables Rt = σ(p˜t+1 + d)/p˜t, Rt = rt+1 + 1 − δ /(1 + g), 26 ˜ b b b b b ˜ s s s s Rt = pt+1(1 + rt )/ pt (1 + gt ) , and Rt = pt+1/ [pt (1 + gt )] denote the growth- adjusted returns to holding stocks, physical capital, bonds and stamps respectively. e e e e In any equilibrium, no-arbitrage implies that R˜ a = R˜ k = R˜ . As in the simple model price bubbles on trees can exist if the real interest is below the real growth rate (R˜ < 1) but not so low that the fundamental assets values in the economy fully absorb aggregate saving (R˜ > R˜ ≥ σ where R˜ denote this lower threshold). If bonds are valued in equilibrium, then no-arbitrage implies that R˜ b = R˜ > R˜ where R˜ b =(1 + rb)/(1 + gb). Thus, the real return to holding stocks, physical capital and bonds respectively is 1 + rb R = (1 + g) . 1 + gb Note that the steady-state real interest rate is given by r = g + rb − gb. Finally, if stamps are valued then no arbitrage implies that R˜ s = R˜ > R˜ where Rs = (1 + g)/(1 + gs). Thus for stamps to be valued, the real interest rate must be below or equal to the real growth rate. Moreover, if stamps are valued, the growth rate of stamps is endogenously determined and equal to gs = gb − rb. To sum up, the following types of equilibria exist. (i) If R˜ = R˜ , then there exist a unique no-bubble equilibrium without non-fundamental asset values. (ii) If R˜ ∈ R˜ , 1 then there exist a continuum of dispersed bubble equilibria with non-fundamental assets values on trees, nominal bonds and stamps. Loosely speaking, these equilibria b m can be indexed by the real value of debt (pt ), the associated lump-sum transfers (τ˜t ), the real value of stamps (p¯s), the price of new public firms p˜ and the asset distribu- t e t,t tion κ.25 (iii) If R˜ = 1 > R˜ , then there exist a continuum of dispersed bubble equilibria with non-fundamental assets values on nominal bonds and stamps. (iv) if R˜ > 1 > R˜ then there exist a unique concentrated bubble equilibrium with government bonds. Before pursuing the quantitative investigation, let us make one additional obser- b b t t t vation. Let mt+1 ≡ pt (1 + gt ) X ∑εt∈E t φb,t(ε ; x0)γ (x0, ε )dκ denote the aggregate demand for government debt inRt, and let gm = m /m − 1 denote the percentage e e t t+1 t b b change between periods t and t − 1. Then, since mt+1 = pt (1 + gt ) by equation (26), we have b e m (1 + gt ) gt = − 1, (1 + πt)(1 + g)

25Note that since lump-sum transfers, the price of new public ﬁrms and the price of stamps depend on the location of the non-fundamental and hence differ across equilibria, the assets distribution and the magnitude of non-fundamental assets also differ across equilibria even though the real interest rate is the same.

27 b b where πt ≡ pt−1/pt − 1 denotes inﬂation. Thus, inﬂation is

b (1 + gt ) πt = m − 1 (1 + gt )(1 + g) b m ≈ gt − g − gt . (32)

m Along a balanced growth path, gt = 0. Thus, inﬂation satisﬁes the quantity theory,

π ≈ gb − g. (33)

However, in the short run (along a transition path), changes in the demand for government debt will cause inﬂation to deviate from the quantity theory.

5 Fitting the Facts?

Regarding all parameters we either use standard values from the literature or calibrate the parameters to match US data. One major caveat is that our model assumes a constant population, whereas the US has typically had considerable population growth. We choose not to take population growth into account, both because it would add complexity (realistic population growth would make the model non-stationary for the next several decades) and because the lower growth rate associated with zero population growth only makes it harder to sustain rational bubbles. However, even if we define the growth rate as the rate of productivity increase, we will realistically be in a regime with r < g. US productivity growth is roughly 2 percent per year for long stretches of time. The real treasury bill rate has been around 1 percent, whereas the rate of return on 10-year government bonds has been about half a percentage point higher.26 Inflation has varied more that real rates. Here, we choose to implement a nominal interest rate of 3.5 percent and a growth rate of nominal bonds of 4 percent, implying a real interest rate of 1.5 percent and an inflation rate of 2 percent in the stationary equilibrium. As we have shown, for any specific r < g there is a continuum of stationary equilibria with different bubble locations. We now compute a dispersed bubble equilibrium with bubbles located on public debt and stamps, but not on stocks. Specifically, we calibrate the public debt to GDP ratio to match US data and let the remaining non- fundamental value take the form of stamps, which is thus the residual variable to be

26Of course, the exact numbers depend on the period that is chosen, but numbers in the vicinity of these occur for many time intervals.

28 determined. The model period is one year. As discussed above we set the coefficient of relative risk aversion µ is 3. The discount factor, β = 0.97, is set to match a wealth/income ratio of 4.5.27 The price mark-up is set to 20 percent (η = 5/6), which is in the mid- range of estimates of US markups.28 The capital share in production, α = .196, is set to match a labor share in income of 2/3. The depreciation rate, δ = 0.06, is set to match an investment to GDP ratio of 0.17. We calibrate the two sources of uninsurable risk as follows. First, the process for uninsurable entrepreneurial rents are governed by three parameters, σ, σo and A1. We conservatively set the survival probability for intermediate goods producers, σ, to 0.98 so that the value of new firms is 2 percent of total stock market value, which is somewhat smaller than the post-war average of 2.3 percent reported by Jovanovic and Rosseau (2005). The fraction σo of entrepreneurs who get the opportunity to make 29 an IPO is set so that the share of private firms is 90.5 percent; σo = 0.00215. These choices help us match the capital-output ratio as well as the stock-market-to-GDP ratio. According to Caselli and Feyrer (2007), the US capital-output ratio is 2.19 and according to data from the World Bank the stock market to GDP ratio in the Unites States has in the last twenty years varied between 0.6 and 1.6,30 with an average of 1.09. In the model, these are 2.22 and 1.05 respectively. Finally, we set the measure of

private firms A1 = 0.015 to match a Gini-coefficient for wealth of 0.816, which is the value reported for the US in 2007 (see D´ıaz–Gimenez,´ Glover and R´ıos-Rull, 2011)31. Second, regarding the process for labor income, we set the transition probabilities between working-age and retirement so that agents spend on average 45 years working followed by 15 years in retirement. The income difference between working and not is determined by labor taxes. The pay-roll tax, τp, is set to 8.5 percent, so that Social Security payments are 4.2 percent of GDP, which is what Wallenius (2013) reports for the US. The labor income tax, τ, is set to 26 percent, so that government expenditure is 19 percent of GDP. Finally, the US public debt to GDP ratio has been roughly two-thirds since 1985 (excluding the run-up during the recent financial crisis). We thus calibrate public debt to be 67 percent of GDP.

27 D´ıaz-Gimenez´ et al (2011) report a non-housing wealth to income ratio of 4 in 1998 and 5 in 2007. 28See for example Basu and Fernald (1997). 29Interpreting private ﬁrms as “small” ﬁrms, this is the number reported by Bartelsman et al. (2004) for industrialized countries. 30The World Bank data refer to Stock Market Capitalization to GDP for United States, series id DDDM01USA156NWDB, World Bank, Global Financial Development. 31The US wealth Gini has increased from 0.79 in 1989 to 0.85 in 2013 (Kuhn and R´ıos-Rull, 2015).

29 The question is: For the chosen parameters, does the model have a solution that hits all the targets? And if so, what does it imply for the residual asset category, stamps? We find that the answer is affirmative, and that the implied stamp to GDP ratio is 0.55. Thus, the total non-fundamental asset to GDP ratio is 1.22. We conclude that our two sources of uninsurable risk suffice to generate a real interest rate below the rate of growth, even after accounting for a large non-fundamental asset price component.

5.1 Savings and wealth

Let us now investigate whether the model also has reasonable implications for savings behavior across states and for the distribution of wealth. Retired agents only face upside chance in this model. It is better to be workers or entrepreneurs. Hence, all retired agents dis-save, gradually running down their assets. As a result 8.4 percent of agents have zero assets, which is close to the 10 percent reported for the US in 2007 (see D´ıaz–Gimenez,´ Glover and R´ıos-Rull, 2011). Workers save to smooth consumption between working life and retirement. But they also face the possibility (albeit small) of becoming an entrepreneur which means higher income. Workers owning less than 32.2 percent of average wealth in the economy save, the remainder does not. Net-savers constitute 81.4 percent of all workers. All entrepreneurs, except those that got the opportunity to make an IPO, take ad- vantage of their temporarily high entrepreneurial income in order to accumulate assets. Entrepreneurs who make an IPO will subsequently be workers or retirees and are always sufﬁciently rich to start dis-saving. What is the wealth distribution that emerges from these ﬂows of saving? Table 1 displays some key moments regarding the wealth distribution in the model and in US data.

30 Table 1 US Model Gini 0.816 0.816 Share of wealth held by bottom 40 0.9 4.2 top 20 83.4 83.7 top 10 71.4 80.4 top 5 60.3 78.6 top 1 33.6 45.7 top 0.1 n.a. 11.2 US data from D´ıaz-Gimenez´ et al (2011)

Given our parsimonious income process, the model captures the data surprisingly well. If anything the wealthiest in the model hold too much wealth, in contrast to most previous heterogenous agent models with uninsurable risk. Moreover, it is in- creasingly clear from empirical studies that ”business wealth and business income are the main drivers of wealth and income inequality” (Kuhn and R´ıos-Rull, 2015, page 72). This suggests that we capture wealth inequality through the correct channel.

5.2 Policy opportunities

To get a rough sense of the how policy affects the magnitude of non-fundamental asset values, we next study the impact of changing the real interest in this economy. Figure 2 displays, for the current parameters, the steady state values of government debt (dashed line) and of the total non-fundamental asset value as a percent of GDP (solid line) for different real interest rates. The ﬁgure calls for several remarks. First, there is a vertical line drawn at a real interest rate of about 0.69 percent. This is the real interest in the no-bubble equilibria (the real interest rate given by R˜ ) without non-fundamental asset values. Below this interest rate, bonds (or stamps) are not valued in equilibrium, and hence neither interest rate policy nor debt policy can affect the real interest rate. Second, as the real interest rate increases, public debt and the aggregate non-fundamental value increase. The magnitude of the non-fundamental value is identical to the value of public debt and stamps when r ≤ g. In computing these dispersed bubble equilibria we have assumed that the non-fundamental value is split between public debt and stamps in the same proportion as in the benchmark calibration when the real interest rate is 1.5 percent. When r > g the non-fundamental value drops to zero.

31 450

400

350

300

Public Debt 250

200 Percent of GDP

150 Non-fundamental

100

0.5 1 1.5 2 2.5 3 3.5 Real interest rate

Figure 2: Interest rates and public debt

Third, note that the relationship between the real interest rate and public debt is close to linear in the relevant ranges below and above the rate of growth, and a one percentage point increase in public debt only requires an increase in the real interest by 1.28 (0.59) basis points. Thus despite market incompleteness, we ﬁnd that the long- run impact of changes in government debt on interest rates is small in the absence of default risk. By comparison, Laubach (2009) estimates that a one percentage point increase in U.S. public debt has been associated with an increase in the real interest by 2 to 3 basis points. To the extent that his estimate is causal, our interpretation would thus be that, when r < g, about a half of the impact on the bond interest rate is due to the net change in supply of savings instruments, and the remainder is due to a change in default (inﬂation) risk. However, as will be clear from the sensitivity analysis below, our model produces estimates in Laubach’s range for fairly modest changes in parameters. Thus, from this exercise we cannot exclude that all of the observed interest rate increase is due to a bigger aggregate supply of assets. Fourth, public debt in the range considered here is backed in the sense that it could potentially be paid down within a reasonable amount of time. In order to get a handle on how debt-repayment might happen and the consequences it would bring, we consider two different cases. In both cases, the labor income tax is raised so as to pay down

32 the debt in ten years, with creditors being reimbursed the full original market value of their claims, principal as well as interest. This requirement of full reimbursement may entail some surprising trade-offs. Case (i): In this first case, we assume that all non-fundamental values vanish. Thus, once the government embarks on a path to pay down the debt, stamps lose all their value. In other words, the new steady-state interest rate drops from 1.5 percent to 0.69 percent (cf Figure 2). The transfers that have been funded through debt creation are reduced as the debt is paid. In this first case, the labor income tax goes up from 26 percent to 40-42 percent during the transition decade, whereupon it returns to its original value. Part of the reason why the labor tax rate increases so much is that the government must keep the interest rate up during the transition in order to accomplish full reimbursement, despite the pressure from savers who know that the interest rate will soon fall. Perversely, this would require wasting rather than transferring some of the taxes. Due to the lower steady-state interest rate, which makes it more difficult to save, and the reduced transfers, the steady-state welfare goes down by an amount corresponding to 1.5 percent of permanent consumption. However, the transitional loss is much greater, and the total loss including the transition exceeds 11 percent of permanent consumption. Case (ii): In the second case, we assume that the value of stamps is increased in proportion to the debt repayment, so as to keep the real interest rate constant at 1.5 percent. In this case, a smaller tax increase is needed to pay the debt, because the capital stock remains on its original path. The transitional tax rates are now in the interval 36-38 percent. Since the steady-state interest rate of 1.5 percent is preferable to the interest rate of 0.69 percent, the ex ante steady-state welfare loss now corresponds to about 0.4 percent of permanent consumption, which is less than a third of the first scenario. However, the transition is still very costly, and the total welfare loss exceeds 6 percent of permanent consumption even in this case. It is possible to decompose the effect of public debt into two components: the transfer component and the interest rate component. For example, in Case (i), the steady- state welfare loss of 1.5 percent is mostly due to the loss of transfers. These account for 1.2 percent, with the loss associated with lower interest rates accounting for 0.3 percent. Still, this computation makes clear that in this calibrated economy there is over-saving at r = 0.69, in the sense defined by Diamond (1967) and analyzed by Davila´ et al (2012). Fifth, since it is desirable to have non-fundamental value being lodged in public debt rather than private bubbles, governments should seek ways in which to move non-fundamental value from private assets to public debt. In principle, it is not diffi-

33 cult to prick bubbles on private assets. According to the model, a tax on the value of the bubbly asset exceeding g − r is enough; then the pre-tax return on the bubble must exceed g in order to yield an after-tax return of r, and hence the bubble is no longer sustainable. Bubble-prevention is thus an additional rationale for taxes on land and real estate as well as on other (low-yielding) stores of value.

5.3 Sensitivity analysis

This section conducts a brief sensitivity analysis. We vary those model features that most strongly affect the demand and supply of assets. On the demand side we focus on the degree of risk aversion, and the sources of risk. On the supply side we focus on our target for the wealth to income ratio, the agents’ ability to borrow, the rate of creation and destruction and our assumption about the stock market. Unless otherwise stated, we present results for re-calibrated versions of the model. In the cases we

consider, recalibration only involves β and A1 and when changing the rate of creation and destruction also σo. Beginning with the demand side, we have run the following experiments. First, we solved the model for different coefficients of risk aversion, µ, letting these take values 2 and 4 in addition to our benchmark of 3. Re-calibrating the model to match a wealth- income ratio of 4.5 and a wealth-Gini of 0.816 requires a higher (lower) discount factor and a larger (smaller) mass of private firms, when agents are less (more) risk averse. The results are very similar. For example, Table 2 shows that when µ = 4, as suggested by estimates in Kimball, Sahm and Shapiro (2009), somewhat lower real interest rates (0.62 percent) are admissible, and the real interest rate is somewhat more sensitive to changes in public debt (a one percentage point increase in public debt requires an increase in the real interest rate by 1.48 and 0.99 basis points when the real interest rate is below or above the rate of growth). Second, the literature on heterogeneous agents and incomplete markets has pri- marily emphasized uninsurable income risk among workers, and in our next experi- ment we add such risk. Labor income risk is typically captured by a stochastic AR(1) process for labor productivity that can can be summarized by the serial correlation coefficient and the variance of the innovation term. Several authors have estimated such processes for the US using data from the PSID, and the results indicate a serial correlation coefficient around 0.95, and a variance in the range 0.015 to 0.06. We adopt the values 0.95 and 0.03 and approximate the productivity process by a discrete Markov process. As Table 2 shows, the results are almost identical to the ones in benchmark case, suggesting that the other two sources of asset demand (uninsurable

34 entrepreneurial rents and life-cycle saving) are driving the low real interest rates. Third, turning to the supply side, we begin by investigating how the magnitude of aggregate supply affects the results. As observed in footnote 27, the US wealth to income ratio has increased in the last two decades. We have re-calibrated the model to match a wealth to income ratio of 4 (as in US data in 1998) and 5 (as in US data in 2007). Again, the results are very similar. Fourth, since Huggett (1993) we know that the stringency of the borrowing constraints matters for the range of admissible real interest rates. We therefore relax agents’ borrowing constraints and add a market for private loans. According to D´ıaz– Gimenez,´ Glover and R´ıos-Rull (2011) the bottom 1 percentile of the wealth distribution had -$79 000 in net wealth in 2007. At the same time they had an income of $38 400 while average income was $83 600. We have chosen to allow agents to borrow up to 94.5 percent (79/83.6) of average income in the model. This implies that the lowest- income agents can borrow 330 percent of their income, which is more than the 206 percent (79/38.4) the data suggest. Again the results are very similar, demonstrating that our benchmark results are not due to overly stringent borrowing constraints. Fifth, as discussed in Section 3.2, the rate of creation and destruction (measured by the value of new ﬁrms entering the stock market) has gradually increased in the post-war period. We have re-calibrated the model, lowering the survival rate σ by two percentage points to 0.96, to match the post 1990 average rate turbulence of 4 percent. Recall that R˜ ≥ σ which implies that the real interest rate is bounded below by σ(1 + g). With a lower survival rate the model thus admits lower real interest rates – in this case down to 0.1 percent. A lower survival rate also makes the real interest rate more sensitive to the level of public debt. When r < g, a one percentage point increase in public debt requires an increase in the real interest by 2 basis points, which is in the range of estimates in Laubach (2009).

35 Table 2: Sensitivity analysis r when r-elasticity2 no-bubbles1 r < g r ≥ g Benchmark 0.69 1.28 0.59 Risk aversion: µ = 2 0.85 0.94 0.14 Risk aversion: µ = 4 0.62 1.47 0.99 Uninsurable labor productivity 0.69 1.28 0.60 wealth Non-fundamental: income = 4 0.93 0.91 0.75 wealth Non-fundamental: income = 5 0.52 1.60 0.45 Borrowing limit: −0.945 × avg. income 0.82 0.94 0.25 Creation/Destruction: σ = 0.96 0.08 2.10 0.75 1. The real interest rate given by R˜ . 2. The long-run increase in the real interest rate (measured in basis points) required to sustain a one percentage point increase in public debt.

Finally, to better understand the importance of our stock market incompleteness assumption, we consider two more versions of the model. In the first version, there is no market for stocks (and hence no market for future projects). In the second, there is an stock market and claims on future projects are implicitly priced through assets that exist today. We implement the first version – maximally incomplete stock markets – by setting the probability, σo, by which each entrepreneur can transform their firm into a public company to zero. This economy is quite similar to our benchmark economy in several respects; in particular there exist stationary bond equilibria with real interest rates both above and below the real growth rate. We implement this second version – maximally complete stock markets – by assuming that that all new firms are off-spring of existing public firms. Thus all firms are public firms. This is essentially the framework of Santos and Woodford (1997). Under this calibration, a stationary equilibrium with positive public debt only exists if the real interest rate is above 6.7 percent. While the introduction of more idiosyncratic risk will entail somewhat lower real interest rates, the model obviously cannot generate r < g without any stock market incompleteness, since this would make the fundamental value of the stocks infinite.

36 5.4 Asset purchases

Until now, the analysis has focused on how interest rate policy (rb) and debt policy (gb) affect the real interest rate and the magnitude of non-fundamental values. Let us now consider the effect of asset purchases by the government. Note first that quantitative easing in the form of issuing new debt to purchase assets has no impact on the real interest rate. Why? Such a policy leaves the government’s as well as the agents’ net wealth unchanged. The government’s additional cost (if r ≥ g, revenue otherwise) of servicing the new debt is exactly offset by the revenue (cost) from holding stocks. Thus, if the government targets a real interest above R (when public debt is valued), then interest rate policy and debt policy suffice. e Suppose on the other hand that the government targets a real interest rate below R. Since public debt is then not valued, both interest rate policy and debt policy are pow-e erless. However, asset purchases funded through taxation can reduce the real interest rate below R. Taxation reduces the agents’ after-tax income, entailing a reduction in private consumptione and saving. Since the government spends all the additional tax revenue to purchase stocks, aggregate saving increases, which in turn reduces the real interest rate. For the sake of illustration, consider our benchmark calibration and suppose that the economy is in a stationary equilibrium with a real interest equal to 0.69 (i.e., R). Suppose that the government would like to reduce the real interest rate to 0.5 percente permanently, and that it, to this end, levies a constant lump-sum tax to purchase assets (stocks or capital). During the transition, the government gradually accumulates assets, and in the new stationary equilibrium the government holds assets equal to 53 percent of GDP and levies additional lump-sum taxes of 0.80 percent of GDP. Even if the government is a net creditor, and the economy has only fundamental asset values, the government can affect the economy’s long-run inflation rate. In order to do so, it merely needs to issue nominal bonds to pay for assets. The return on the assets is then used to pay interest on the bonds. In our model with a single real interest rate, this operation has no effect on the government’s surplus, and it keeps agents’ net wealth unchanged. Along a balanced growth path, inflation keeps obeying π = gb − g.

6 Other Related Literature

There are two complementary theoretical literatures on rational bubbles. The first emphasizes that firms may be financially constrained due to the limited pledgeability of future returns or the lack of collateral; see Woodford (1990), Holmstrom¨ and Tirole

37 (1998), Kiyotaki and Moore (2002, 2003), Kocherlakota (2011), Farhi and Tirole (2012), Giglio and Severo (2012), and Mart´ın and Ventura (2012). In these models, unlike ours, financial frictions keep investments below their first-best level.32 The marginal rate of return to capital thus exceeds the rate of return on marginal savings, which in turn may be low enough to permit bubbles even if the economy satisfies Abel et al’s (1989) dynamic efficiency criterion. This approach emphasizes the precautionary savings behavior of firms for future investment purposes rather than of individuals for future consumption purposes. By doing so, it admits a more nuanced role for bubbles. In particular, bubbles and Ponzi schemes need not crowd out investment, as it does in the classical literature (as well as in our model). By providing an instrument that financial constrained firms can use for the purpose of saving, non-fundamental asset values may even be promoting investment.33 While this literature addresses the dynamic efficiency critique, it has remained vulnerable to the SW-critique: Either it has invoked overlapping generations, like Farhi and Tirole (2012), or it has heavily re- stricted the set of assets or their trade, as in Bewley (1980,1983) and Woodford (1990). We conjecture that our notion of mortal alienable rents can provide an escape also in these environments – a way to reconcile dynastic agents with reasonable contracting frictions without eliminating the bubbles.34 The second related literature, represented by Kehoe and Levine (1993), Kocher- lakota (1996), Alvarez and Jerman (2000), and Hellwig and Lorenzoni (2009), is concerned with studying the maximum levels of debt that can be sustained by agents who have limited commitment abilities. The central idea in Hellwig and Lorenzoni (2009) is that the threat of not being able to borrow in the future can suffice to sustain repayment discipline today even if the only punishment is to be prevented from borrowing in the future. Governments will refrain from defaulting on debts that are owed to foreigners whenever the interest rate is below the rate of growth, because it is a harsh punishment not to be able to engage in future borrowing under such favor- able conditions. Hellwig and Lorenzoni find that the conditions under which public debt is sustainable are exactly the same as the conditions under which bubbles are sustainable, namely r ≤ g. This is an intriguing relationship between the debt being non-fundamental and the government’s promises being credible, and it responds at

32Our choice not to consider ﬁnancial frictions should not be interpreted as a denial of the empirical relevance such frictions. 33If bubbles are attached to productive assets, private bubbles may also be preferable to public Ponzi- schemes in this sort of a model; see Olivier (2000) and Ventura (2003) for early models in which bubbles are attached to investment and entrepreneurship opportunities respectively and Farhi and Tirole (2012) for a synthesis. 34So far, this literature has had modest ambitions by way of aggregate quantitative analysis. Thus, another challenge is to integrate its insights into a workhorse macro-model.

38 least partially to the objection that large public debts may be unsustainable because of strategic default concerns.35 Whenever r < g strategic default is not a concern.36 Our contribution complements this line of work by provide a theoretical analysis of why and when equilibrium interest rates on government debt can become so low in the first place. There is also a large empirical literature on rational bubbles on the prices of indefinitely lived assets. Giglio, Maggiori, and Stroebel (forthcoming) provide a recent overview, with an emphasis on the statistical tests for rational bubbles in property markets. They argue that the literature is inconclusive, largely because the findings rely on indirect tests. More importantly, they argue that there are markets in which we might directly measure the magnitude of rational bubbles by comparing the price of assets with indefinite duration to the price of similar assets with finite duration. Their empirical analysis compares the price of residential property in the form of freeholds (full ownership) and lease-holds (finite-duration ownership) in UK and Singapore. Since they focus on very long (700-999 years) leases and carefully control for differences in characteristics, any price difference should reflect a bubble on the freeholds. Their point estimate for England and Wales is that the price difference is around 0.001, with a standard error of 0.005. Giglio, Maggiori, and Stroebel conclude that there are no economically significant rational bubbles in these residential property markets. How- ever, this conclusion rests on the assumption that the probability of enfranchisement is so small that it can be neglected, which appears debatable with such long horizons.37

35In our closed economy setting, the debt is owed to (a subset of the) domestic citizens, and it is not clear that the government would want to default on these citizens even if it could. Thus, for governments that borrow domestically, it might be credible to borrow also at interest rates r > g. 36Rather, the relevant concern is the possibility that there is a run on the public debt; investors may coordinate on an equilibrium in which roll-over is denied. In that case, the issue of backing – the quality of the tax system – is crucial. 37Looking at leases with long horizons ensures that only bubble values will matter at the horizon, but the downside is that we can be less sure that the lease will in fact have ﬁnite duration. Consider a typical case, in which the lease has about 900 years until expiration. Suppose there is a constant yearly probability f > 0 that the lease-holder will be enfranchised for free. (We may neglect the possibility that the owner-occupant can be disenfranchised again, since this risk is also present for current freeholders.) By the expiration date, there will then have been enfranchisement with probability 1 − (1 − f )900. For example, when f = 0.008 the enfranchisement probability is larger than 0.999. Under the condition that bubble values do not depend on the property’s lease-history and that we may abstract from risk- aversion, the bubble on lease-holds should thus be above 99.9 percent of the bubble on freeholds. In this case, the test of Giglio, Maggiori and Stroebel would fail to reliably detect the difference in price even if the price were entirely a bubble. If we reduce f to 0.002, the enfranchisement probability drops to 0.835. With a standard error of 0.005, the test detects all bubbles b satisfying (1 − 0.835)b > 0.01, that is, it detects all bubbles exceeding 6 percent of the price. Giglio, Maggiori and Stroebel argue that f is likely to be small because enfranchisement would represent a large wealth transfer that British courts are loath to condone. However, if the authors are right that there is no bubble, the wealth transfer is actually negligible. Even with a conservatively low net discount rate of 0.01, the transfer would represent a fraction of only 0.99900 ≈ 0.0001 of the current price.

39 Regardless of this qualiﬁcation, the methodology of Giglio, Maggiori, and Stroebel takes the measurement of rational bubbles a large step forward, and we are optimistic that it will spur additional progress on the empirical front.

7 Conclusion

One line of research, usually building on OLG models, has argued that rational bubbles and Ponzi-schemes are plausible phenomena. Another line of research, usually building on dynastic models, has argued that they are implausible. By many accounts, the latter line of research has come to dominate economic policy debates over the past two decades. There is a widespread view that asset price bubbles are signs of irrationality, that they are undesirable, and that they are bound to burst eventually. There is also a widespread view that current deficits in public finances must be repaid through future surpluses and that fiscal responsibility means holding down public debt. The main purpose of this paper is to show that rational bubbles and Ponzi-schemes plausibly occur in dynastic economies too. Levels of asset market incompleteness that correspond to those observed in reality prevent individuals from eroding rational bubbles and Ponzi-schemes through arbitrage. Therefore, at least for countries with well- functioning tax systems, public debt really is as cheap as it seems, if not cheaper. It can be rolled over forever, and larger public debt helps to keep real interest rates at a level that supports adequate private precautionary saving. If possible, private bubbles should be prevented through appropriate capital taxation, so as to leave even more room for public debt. Obviously, our analysis is far from final. We see it as an early attempt to introduce a realistic asset structure in dynastic economies. The most obvious weakness of the current model is that it only admits one interest rate. In future work, we hope to generalize the model to include aggregate shocks and to re-calibrate the model so as to have low interest rates on debt while at the same time have a high return to equity, allowing the return to capital to exceed the rate of growth – i.e., to replicate the dynamic efficiency that is observed empirically. This is a technical challenge as much as a conceptual challenge, since it is difficult to solve portfolio problems with many assets in this kind of heterogeneous agents setting. We also hope to exploit this framework to provide a theory of economic crises based on bubble movements across asset classes. Endogenizing such bubble movements is perhaps the greatest challenge of all, as it requires both aggregate shocks and a more refined solution concept. Intu-

40 itively, we think that uncertainty about private bubble creation would cause migration of savings towards public debt, whereas uncertainty about the backing of public debt would cause migration of savings away from public debt. Under a passive fiscal re- sponse, our model indicates that the former case would be associated with deflation and the latter with inflation. With nominal price rigidity, the deflation case might instead be associated with a contraction in economic activity due to “lack of demand” as goods prices are too high to clear the market. A version of the model with multiple interest rates and nominal rigidities might thus be consistent with the notion that there has been a scarcity of safe assets during the recent Great Recession. For closely related ideas, see Caballero, Farhi, and Gourinchas (2008) and Caballero and Farhi (2016). There are also several questions that could be addressed by more straightforward extensions of our model. For example, what is the optimal level of public debt? To address this question properly, one should follow the example of Floden´ (2001) and admit elastic labor supply. Taking into account the fact that income taxes are distor- tionary, the analysis should then optimize over all the fiscal instruments. Other questions are: What are the pros and cons of nominal debt as opposed to real? What are the reasons for and implications of having bonds with different maturities? Finally, there is the eternal question of money. How should we think about the relationship between currency and bonds in this kind of framework?

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