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Credit Growth, Rational Bubbles and Economic Efficiency.

Introduction

The recent crisis has shed light on two of the most important flaws of traditional banking regulation: the too narrow delimitation of the perimeter of the regulated banks' activity and the lack of mechanisms to prevent and mitigate systemic crises. As a reaction, new regulatory rules covering shadow banking as well as the implementation of macroprudential policies have emerged as remedies to such limitations. These innovations are important challenges from a policy perspective that require a sound theoretical understanding of the issues at stake.

This paper argues that the emergence of bubbles plays a key part in the endogenous building of financial imbalances and should be part of the theoretical toolkit on which regulation is based. Consequently, the understanding of the link between bubbles and crises should shed light on the advantages and drawbacks of some of the newly proposed regulatory measures and particularly on macroprudential policy. Indeed, as emphasized by Benes et al. (2014, p. 4) "the critical macroprudential policy trade-off is between reducing the risks of very costly financial crises and minimizing the costs of macroprudential policies during normal times."

There is nowadays growing empirical evidence on the relationship between excessive credit growth and financial fragility, so that it is tempting to focus on credit and disregard bubbles in order to understand the endogenous building of systemic risk. This would be erroneous in so far as an increase in the volume of credit could be the result of an increase in the (solvent) demand for loans or in its supply, which results from changes in some exogenous variables. Per se, the evolution of credit tells us nothing about the driving forces behind it. The existence of bubbles, instead, offers a rationale for the demand of credit, and the bubble on the equilibrium price of the corresponding asset depends upon the supply of credit.

This is the reason why the analysis of bubbles is particularly interesting. Indeed, it provides a framework where both the demand for credit and the supply for credit, often guaranteed by the price of the asset, increase as asset prices increase. From the policy perspective, "Research on asset prices, credit, and intermediation should help to identify risks and inform decisions about the costs and benefits from a possible regulatory or monetary policy decision attempting to deal with a potential asset price bubble" (Kohn 2009).

Obviously, the existence of bubbles is not the only approach to explain credit cycles and systemic crises. Indeed, alternative approaches, such as the behavioral approach to loan screening (Berger and Udell 2004), business-cycle-related changes in credit standards (Lown and Morgan 2006; Maddaloni and Peydro 2011) and credit cycles (Kiyotaki and Moore 1997; Dell'Ariccia and Marquez 2006), provide justifications that are fully in line with the existing empirical evidence.

There are many ways to model bubbles, some of which we briefly describe below. There is a divide in the literature between the rational and the behavioral approach to bubbles modeling. The empirical evidence is consistent with both, although experimental economics has shown the limits of rational behavior as bubbles could develop in experiments even with a finite number of trading periods (Smith et al. 1988). Nevertheless, when the ultimate objective is to try to draw some lessons regarding macroprudential policy, the behavioral approach has a number of limitations that makes it impracticable. First, although it is easy to reject rational behavior, it is more difficult to replace it with a specific pattern of limited rationality or irrational behavior. Second, even if a specific irrational behavior is identified as representative of consumers preferences, it is not clear how to construct a measure of welfare that takes into account the irrational behavior. Since, in order to build a strong foundation for macroprudential policy, it is essential to have a welfare function to compare the efficiency of the different policies, we will focus here exclusively on the theory of rational bubbles and its implications on macroprudential policy.

The structure of the paper is the following: in "Credit Growth, Bubbles and Crises: Empirical Evidence" section, we examine the existing empirical literature that links financial crises to credit expansion, leverage and bubbles. In "Bubbles and Crises" section, we survey the main theoretical results on bubbles. "The Theory of Rational Bubbles" section is then devoted to the role bubbles may have in a market characterized by financial frictions. "Bubbles and Credit Rationing" section considers the impact of financial intermediation, and "Bubbles and Financial Intermediation" section the macroprudential implications of the existence of bubbles is explored. "Macroprudential Policy Implications: Throughthe Bubble Lens" section concludes.

Credit Growth, Bubbles and Crises: Empirical Evidence

Since Kindleberger (1978) classical book, it is well known that financial crises are intertwined with the increase in debt, credit booms and the soaring of asset prices. Today's accumulation of empirical evidence allows us to have a more precise view of the impact of these different factors in the building of risk that precedes a banking crisis. These empirical results are interesting per se, but they also help in building theoretical models that are in line with the facts.

Excessive Credit Growth

The abundance of cross section and time series around financial crises raises a point that is preliminary to the view of empirical results and that is whether there has been a structural change in the way financial systems operate, either due to the modernization of financial systems, (economic and technological development, financial innovations, etc.) or to the use of fiat money. While in the 1960s Friedman and Schwartz (1963) were able to identify the link between monetary aggregates and the building of risks leading to the crisis of 29, the approach may be insufficient to explain contemporary crises.

The criticism of the monetarist approach started with Bernanke (1983) and has led to the a major debate in monetary theory between the money and the credit view. Broadly speaking, the former could be associated with the aggregation of different financial markets in a unique market for loanable funds, only the money market matters, and interest rates are the main determinant of macroeconomic aggregates. The credit view, instead, posits credit as a key variable that affects economic activity.

The empirical evidence obtained by Schularick and Taylor (2012) shows the relevance of the credit view. In their study of money, credit and macroeconomic indicators based on a historical dataset covering 14 countries over the years 1870-2008, the authors show, first, that, prior to 1950, the stability of these series would be consistent with the monetarist view and would not suggest any need to analyze broader credit aggregates. Credit is just the asset item in the balance sheets, and money is the liability counterpart. Second, they establish that the post- 1945 era was characterized by bank loans and assets increasing relative to GDP, while broad money relative to GDP remained stable. As credit began to be disengaged from broad money, it could grow rapidly, via increased bank leverage.

Still, the evolution of financial systems may be even more complex if the allocation of credit across sectors has changed, either as a disruption or as a secular trend. Jorda et al. (2014) document the "financialization" of the economy and how it relates to a change in the role of the allocation of capital, with a rising share of real estate lending (i.e., bank loans secured against real estate) in total bank credit and the declining share of unsecured credit to businesses and households. They also argue that "Mortgage lending booms were only loosely associated with financial crisis risks before world war II, but real estate credit has become a more important predictor of impeding financial fragility in the postwar era." In addition, they show that "the magnitude and structure of credit booms have important consequences for business-cycle dynamics."

Such a change in the portfolio of financial intermediaries does not imply that the aggregate volume of credit increases. In a companion paper, Jorda et al. (2011) proceed to analyze the link between deregulation and credit growth. They argue that a strong and sustained credit boom cannot be financed with local increase in deposits and wealth (especially if not driven by very strong fundamentals); foreign liquidity, or liquidity stemming from expansive monetary policy or financial innovation (e.g., securitization), needs to be present and to interact with the credit cycles. Instead in a globalized economy with free capital mobility, credit cycles and foreign capital flows have the potential to reinforce each other more strongly than otherwise. The importance of foreign inflows is also emphasized in Dell' Ariccia et al. (2012) who show that, during a boom, imbalances build and the current account deteriorates by slightly more than 1 % points of GDP per year, so that a corresponding increase in net foreign liabilities, which includes banks funding by foreign sources, is required.

Credit Growth and Financial Stability

In the analysis of the link between credit growth and financial stability, an accurate definition of financial crises and excessive credit growth has to be put forward. (1) In spite of the unavoidable imprecision in the identification of both financial crises and excessive credit growth, there is a clear consensus that rapid expansion of credit is associated with the subsequent higher likelihood of a recession (Schularick and Taylor 2012; Jorda et al. 2013; Drehmann and Juselius 2014) and that credit growth constitutes a predictor of financial crises (Jorda et al. 2015). Also, besides the probability of a crisis, the cost of the crisis is also a key issue.

Reinhart and Rogoff (2009) reach a similar conclusion, although they emphasize the role of debt, both public and private, as well as the role of external debt, which is, itself, correlated with credit growth.

In spite of the convergence in the results, there are two key difficulties in using these results as early warnings and as a guide for macroprudential policy.

First, while credit booms are clearly a critical ex ante correlate of financial crises, the analysis cannot make strong causal inferences on the net effects of credit. Potential explanations for these effects include financial accelerator effects, debt-overhang problems or the impact of shifts in expectations.

Second, and possibly more important, although the probability of a crisis increases after a credit boom, the results do not imply that the majority of credit booms end up in a crisis. According to Dell'Ariccia et al. (2012) analysis, since 1970 two-thirds of the credit boom episodes did not generate a financial crisis. Conversely, as Laeven and Valencia (2012) report, "Out of 129 banking crises episodes for which credit data are available, 45 episodes (or about one in three) were preceded by a credit boom" (p. 10).

Because credit volume is an endogenous variable, implementing a macroprudential policy requires to analyze the factors that increase the demand and supply of credit. As we have mentioned, the supply side is justified by the growing imbalances reported by Dell'Ariccia et al. and the increase in foreign capital inflows it implies. The issue of demand is a more complex one. It may be driven by lower interest rates, high rates of growth that required additional investment, or high levels of innovations and technical progress. All these factors would be justified in terms of efficient investment. Still, the empirical evidence points out, rather, at an increase in credit that concentrates in real estate. Consequently, the introduction of bubbles as a possible justification for the increase in the demand for credit and the prices of real estate would be a possible explanation. If so, the existence of bubbles should also predict banking crises. In other words, excessive credit growth may be traced back to the financing of a bubble.

Bubbles and Crises

Brunnermeier et al. (2009, p. 32) in the Geneva Report on the analysis of the crisis argue that "Most financial crises are preceded by asset price bubbles" Jorda et al. (2015) show that "when fueled by credit booms, asset price bubbles increase financial crisis risks; upon collapse they tend to be followed by deeper recessions and slower recoveries." Credit-financed housing price bubbles have emerged as a particularly dangerous phenomenon. Carvalho et al. (2012) also point out that the ratio of wealth to GDP increased prior to the crisis, which is consistent with the existence of a bubble affecting the ratio's numerator but not the denominator.

The analysis of bubbles is controversial for several reasons. First of all, it is difficult to identify them, as repeatedly pointed by Federal Reserve Governors Alan Greenspan and Ben Bernanke. Indeed, from a theoretical perspective it is easy to define a bubble as the difference between the market price and the value of a fundamental. Still, the fundamentals depend upon expected future cash flows, the risk and liquidity premia, and other variables particularly difficult to predict.

From the empirical research perspective, the first challenge is to identify a bubble, which is a daunting task. Different methodologies could be applied. One option is to define housing price booms as deviations of real house prices above some specified threshold relative to trend (Borio and Lowe 2002; Detken and Smets 2004; Goodhart and Hofmann 2008). Alternatively, Bordo and Jeanne (2002) focus on deviations from the long-run fair value. Another way to identify bubbles would be based on the peak-trough difference (Helbling 2005; Helbling and Terrones 2003; Claessens et al. 2008). Finally, it is also possible to test econometrically if a series of variables are in a regime characterized by an explosive behavior or not (Phillips et al. 2011, 2015) and associate the explosive behavior to the emergence of a bubble or exuberant state (Anundsen et al. 2014). These methodologies can be combined, as Jorda et al. (2015) do, in order to avoid defining bubbles that might never burst, which may lead to classify increases in real estate prices that are due to changes in the fundamentals as bubbles. For this reason, they combine their definition with the requirement that at some point a large price correction occurs.

In spite of the drawbacks of such a multiplicity of definitions, the empirical analysis establishes the existence of a correlation between the bursting of bubbles and financial crisis. Thus, Claessens et al. (2011) and Borio (2012) suggest that the financial cycle can be described parsimoniously in terms of credit and property prices, with rapid growth in these variables providing an important early warning indicator of potential future financial crises.

The Theory of Rational Bubbles

General Properties

The equilibrium price of a bubble, [p.sub.t], is directly related to the expectations agents in the economy hold. Under rational expectations, this potential source of indeterminacy is eliminated as it imposes a strong condition of internal consistency. When risk neutral agents have access to several assets, the expected returns they obtain from investing in any asset has to yield the prevalent interest rate [r.sub.t+1]. This condition, usually referred to as the no-arbitrage condition, leads to a recursive equation involving the future expected price, [p.sub.t+1], and expected dividend [d.sub.t+1]:

[p.sub.t] = [E.sub.t] ([p.sub.t+1] + [d.sub.t+1])/1 + [r.sub.t+1] (1)

By iterating this equation and assuming, for the sake of simplicity, constant interest rates, we obtain a decomposition up to T.

[[p.sub.t] = [E.sub.t] [[[tau] = T-t.summation over ([tau]-t)] [d.sub.t+1]/[(1 + r).sup.[tau]] + [E.sub.t] [[p.sub.T]/[(1 + r).sup.T-t]]]

When T tends to infinity, the first term is the fundamental value of the asset, while the second, if strictly positive, constitutes the value of the bubble. The decomposition allows to see that, for finite assets or finite horizons, by backwards induction, [p.sub.[tau]] = 0 for [tau] [greater than or equal to] T, and a rational expectations bubble cannot exist in a perfect capital market with homogeneous agents (see Allen et al. 1993). For infinite assets, the value of the asset corresponds to its fundamental if and only if the so-called tranversality condition [lim.sub.T[right arrow][infinity]] [E.sub.t][[p.sub.T]/[(1+r).sup.T-t] = 0 is satisfied.

When the tranversality condition is not met, a multiplicity of solutions involving positive bubbles exists. The case of a pure bubble, where, because [d.sub.t+1] = 0, the fundamental value is zero, suffices to illustrate this point. In the absence of additional constraints, there is an infinity of non-random steady-state solutions that depend upon an arbitrary initial price [p.sub.0]: [p.sub.t] = [p.sub.0][(1 + r).sup.t]. When the interest rate is zero, the price is constant, when it is negative, the price decreases toward zero, and when interest rates are positive, the price would tend to infinity, provided the rate of growth of the economy is high enough to sustain it.

The rational expectations equilibrium will be constrained by the feasibility and free disposal. First, if the interest rate r is larger than the rate of growth of the economy, a bubble cannot exist because of the feasibility constraint being violated at some point. (Bubbles in zero supply will not face this constraint.) Also, there is no negative bubble, and a negative price would contradict the free disposal (or limited liability) assumption. (2)

The term [E.sub.t]([p.sub.t+1] + [d.sub.t+1]) in Eq. (1) could depend upon the probability of the bubble bursting. This is to be noted because of its potential effects on the bubble price and on systemic risk. As noted by Blanchard and Watson (1982), when the bubble either bursts (with probability 1 - [pi]), or has a price increase with probability [pi], a non-bursting bubble has to grow at a rate 1 + r/[pi] - 1, in order to satisfy the no-arbitrage condition and yield a net expected return r.

Obviously, beyond the no-arbitrage condition, in a complete model, the price will be determined by households, entrepreneurs and financiers.

With an infinite time horizon, two set ups are possible, one with infinitely lived agents and the other, of overlapping generations, where young generations can only trade with old generations. More specifically, as in Diamond's (1965) celebrated paper, the young generation works and decides on its saving so as to consume when old. The equilibrium levels interest rate, [r.sub.t+1], capital, [k.sub.t], and bubble price, [p.sub.t], result from agents maximizations. From these, it is possible to derive the agents' consumption, how they transfer their savings across periods and their equilibrium wages if labor is introduced.

Dynamic Inefficiency in OLG Models

Diamond shows that a competitive equilibrium with production may not be efficient as bubbles may crowd out efficient investment and lead to dynamic inefficiency, "despite the absence of all the usual sources of inefficiency" (p. 1126). Analyzing this issue, Tirole (1985) shows that no rational bubbles exist when interest rates are larger than the rate of growth. Consequently, bubbles can only emerge in dynamically inefficient economies, where in the absence of bubbles, an equilibrium with low interest rates may exist in such a way that, in order to transfer resources across generations, the amount of capital accumulated is so large that the marginal investment exceeds the marginal income it produces. (3) Such an investment lowers the resources available for consumption and is dynamically inefficient as the shift of resources from one period to another generates gains from trade across all time periods. In this situation, bubbles can be both attractive to investors and feasible.

The model can be enriched, so as to take into account stochastic bubbles, as in Weil (1987), particularly if the bubble is determined itself by exogenous stochastic factors, such as technology, credit or as in Froot and Obstfeld (1991) who explore the formation of bubbles in the stock market depending on dividend distribution. When a stochastic equilibrium is considered, the allocation depends upon a state of nature that may consist of productivity and liquidity shocks. In this case, how expectations are formed, will, in a rational expectation context, depend upon the probability distribution of the state of nature, which may, for instance, be stationary or conditional on the realized state of nature, as in the case of a random walk.

Still, these classical results are obtained under the assumption of a perfect capital market, so that, if the production function is f([k.sub.t]), a function of the capital input [k.sub.t], entrepreneurs will set their marginal product equal to the interest rate:

df([k.sub.t])/d[k.sub.t] = 1 + [r.sub.t+1] (2)

In its simplest expression, the existence of a riskless asset in sufficient large supply will determine [r.sub.t+1]. Alternatively, the riskless asset may be in limited supply and the savings allocated to the riskless asset, to productive investment and to the bubble will determine the equilibrium interest rate.

If, instead, entrepreneurs are constraint on their access to credit, for instance because their loans have to be fully collateralized, then:

[mathematical expression not reproducible] (3)

where [bar.[k.sub.t]] may depend upon the firms' collateral and its future pledgeable income. In this case, there is a shadow price for credit and the interest rate does not convey. The economy may then alternate between firms being credit constrained or not, depending on the aggregate level of savings. When this is the case, interest rates are disconnected from capital accumulation and low interest rates need not lead to capital overaccumulation.

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When entrepreneurs are rationed, in order to close the model, a third equation is required, and it will be the equilibrium in the intraperiod market for credit that connects the savings of one generation or one type of agent to the investment or consumption needs of another. The demand for credit D(r, [bar.[k.sub.t]], [p.sub.t]) or the effective demand for credit when condition (3) holds and firms are rationed will come from entrepreneurs and from leveraged buyers of the bubble. The supply of credit S(r, W), which depends upon interest rates and wages, will be the result of the young generation's savings for their future consumption or from financial intermediaries that, in turn, are financed by depositors. In equilibrium,

D(r, [bar.[k.sub.t]], [p.sub.t]) = S(r, W, [p.sub.t]) (4)

While the limit on credit affects only the effective demand for credit and the young generation's wealth (its wage W) affect only the supply, interest rates affect both and so does the current price of the bubble, which may lead some agents to switch from being a net saver to being a net investor.

Bubbles and Credit Rationing

While in a frictionless market bubbles are associated with dynamic inefficiency, this is not necessarily the case when they play a role in mitigating market imperfections. Woodford (1990) and Azariadis and Smith (1993) showed that financial frictions could relax the conditions for the existence of rational bubbles. In the presence of financial frictions, the economy can be dynamically efficient, and at the same time, the interest rate on borrowed funds can be lower than the growth rate of the economy, as it appears in the above Eq. (4) . In other words, the shadow cost of funds exceeds the interest rate, yet the latter may be lower than the growth rate. Interest rates may be low because the (collateralized) demand for funds is low and, if anything, there is underinvestment in the productive sector. As a result, bubbles are possible even when the economy is dynamically efficient.

The introduction of financial frictions will affect the three equations that determine the OLG equilibrium: the no-arbitrage condition, the actual marginal cost of funds, interest rate or collateral constraint and the equality of supply and demand in the market for funds.

Thus, in the presence of moral hazard, whether related to the firms' project choice (Holmstom and Tirole 1997, 1998) or to the possibility of payment renegotiation (Hart and Moore 1994), when loans can only be granted against collateral, bubbles could be an additional source of collateral for firms and will improve the efficiency of allocation by providing entrepreneurs with additional funding capability. The existence of a bubble in the assets held by entrepreneurs will supplement the collateral base on the firm's future output or tangible assets and will increase credit and economic activity. The other side of the coin is, first, that the bursting of a bubble may originate a banking crisis and, second, that too large a bubble might deprive entrepreneurs from the benefit of additional collateral if credit is mainly used to cancel the previous period debt commitments, leading to a crowding-out effect. Consequently, there may be an optimal level for the bubble that maximizes long-run output and consumption, and this characteristic will, of course, have important implications for macroprudential policy.

For bubbles to have a value as a source of future collateral, it is necessary that the corresponding asset is held by entrepreneurs when they need to borrow. This can lead to different modeling approaches.

In Farhi and Tirole (2012), there are no consumers and entrepreneurs live for three periods. They are born with an endowment they save; then, they invest when middle aged, and because production takes one period, they obtain the return on the production when old, which allow them to repay their loan and to consume. There is no risk in the production process or in the endowment, and the growth rate is (normalized to) zero. Still, because of moral hazard, the project cash flows are only partially pledgeable.

When entrepreneurs are young, they have a choice between three saving possibilities: (1) invest in outside liquidity, akin to a storage facility, that yields 1 in the next period, but is in limited supply l, (2) buy the bubble or (3) buy claims on the future production that are currently issued by middle-aged entrepreneurs. These three channels will provide self financed credit to the entrepreneur and will be part of S(r, W) in the above Eq. (5). The demand for funds D(r, [bar.[k.sub.t]], [p.sub.t]) will be determined as a corner solution by the credit rationing condition (3). In equilibrium, the three alternatives channels for saving will yield the same return, because of the noarbitrage condition generalized to two interest rates, so that the interest rate in (1) is both the return on the outside liquidity and on claims on future production.

The supply l of the storage technology as well as the supply of bubbles is exogenously given and constitute outside liquidity, as it is unrelated to the future production. Inside liquidity is instead generated by entrepreneurs and, therefore, depends upon their current investment, which determines their future output. Because access to funds depends upon the pledgeable collateral, a higher outside liquidity implies a higher level of production, which, in turn, implies higher collateral and a higher volume of loans. Consequently, bubbles may crowd in investment.

The equilibrium will then determine the interest rate in the economy through the equilibrium in the market for funds, as middle-aged entrepreneurs invest borrowing from young ones. The no-arbitrage condition (1) holds here for the three assets available, and the collateral constraint allows to close the model.

Several types of equilibrium can then emerge.

First, there is a bubble-free equilibrium. This equilibrium has a steady state (4) that implies a positive interest rate and any bubble-free competitive equilibrium converges to the steady state.

A second equilibrium exists with bubbles, where there is a unique steady-state equilibrium with a constant bubble and a zero interest rate.

The comparative statics shows that, in the bubbly steady state, variations in the liquidity storage facility are compensated one by one by the equilibrium price of the bubble. In other words, the total outside liquidity is constant and there is no effect on interest rates, nor on investment.

As the model allows for a bubbly and a bubble-free equilibrium, the introduction of a sunspot equilibrium allows to close the model in a simple way. The switch from the bubbly equilibrium to the bubble-free one is interpreted as a bursting of the bubble, so that the return on the bubble is stochastic and leads to high return if the bubble does not burst and to zero if it burst.

Martin and Ventura (2016) also show that bubbles can be efficient in collateral constrained economies but point out that they can lead to inefficient outcomes. They consider a world of risk neutral agents with fixed intertemporal discount rates and a capital market where entrepreneurs buy the bubble when young, use it as collateral to borrow from savers in the intraperiod financial market, sell it to the next young generation and repay their debts.

They point out that entrepreneurs are able to issue new bubbles, which apparently would contradict Diba and Grossman (1988) who argue that the creation of new bubbles after the first period is impossible. This is the case because if the bubble is not priced at time t = 1, it is because it has a zero expected value. Still, under limited liability, bubbles cannot be negative, and consequently, their value is zero. Now, if a bubble becomes zero at some point, under rational expectations, it should be zero forever. Martin and Ventura (2016) observe this is no the case for firms which securities were not initially traded and come to live at some ulterior date.

The financial market friction stems from the fact that the net future profit accruing to entrepreneurs can only be used as collateral with a discount, while entrepreneurs are able to pledge the full future value of the bubbles they hold in order to get credit from consumers. As a consequence, entrepreneurs that have purchased bubbles have a better access to credit. The implication is, as in Farhi and Tirole (2012), that bubbles, capital stock and interest rates are jointly determined. If the capital stock is low, the interest rate is above the discount rate and young consumers save by investing in the firms where collateral is abundant rather than consume. In this case, collateral is abundant and the capital stock is determined by the supply of funds. If instead the capital stock is high, the interest rate equals the discount rate and young savers consume part of their income. In this case, collateral is scarce and the capital stock is determined by the demand for funds.

In both instances, the positive effect of collateral (crowding in) is therefore to increase the amount of pledgeable assets the entrepreneurs hold and, consequently, increase the amount of credit available to them. While this is efficient when credit is rationed because of limited collateral, in Martin and Ventura it has a negative impact once the bubble becomes excessively large. This is the case because entrepreneurs will have to allocate a part of the credit to the repayment of the bubble. There will then be crowding out because the excess credit will divert some of the resources of future generations away from investment. This implies an excessive bubble may be inefficient even if it there is no risk it burst. Thus, in the Martin and Ventura framework, an "optimal" bubbles size can be determined. As a consequence, there is a role for macroprudential policy in targeting the right bubble size and avoiding both crowding in and crowding out.

When the risk of a bubble bursting exists, the Blanchard-Watson argument applies and, conditional on the bubble not bursting, riskier bubbles become larger and eventually lead to more severe crowding-out effects.

Bubbles and Financial Intermediation

The introduction of banks in the model introduces an additional layer of complexity. First, because of the cost of financial intermediation, deposit rates differ from lending rates and, therefore, it has to be specified whether it is the deposit or the lending rate that enters the no-arbitrage Eq. (1). Second, it has to specify financial intermediaries access to deposit and equity that this will determine S(r, W). In particular, banks that face capital regulation may have to limit their supply of credit and their demand for deposits.

The Specificity of a Banking Framework

The existence of banks implicitly presupposes that these financial intermediaries have a role in improving the allocation of funds, whether as delegated monitoring (Diamond 1984) or by reducing asymmetric information in some other way. Whatever the raison d'etre for the emergence of financial intermediation, it implies, in addition to the existence of a spread between deposit and lending rates, a high impact of banks' bankruptcy in terms of the allocation of funds to firms. This can be made explicit by combining the framework of OLGs with a model that justifies the existence of banks, such as Holmstom and Tirole (1997). If a bank equity is insufficient to offer a sufficient buffer, then the bubble bursting will wipe out its capital and firms' access to funding will be reduced. When this affected a significant number of banks or a systemic bank, it will lead to a systemic crisis, an important reason to implement a macroprudential policy.

Because of the cost of a systemic crisis, the Blanchard-Watson argument has an important implication. Indeed, if the bubble does not burst, banks holding a bubble will obtain larger profits and expand both their capital and their supply of credit. This will have a positive effect if firms were credit rationed, but will also increase systemic risk.

Banks' Incentives to Hold the Bubbly Asset

Aoki and Nikolov (2015) consider the existence of bubbles in a banking economy and their role in generating banking crises. Instead of the overlapping generation model, they consider infinitely lived agents and explore the equilibrium when both banks and households are able to invest in the bubble. This is a particularly interesting point as it could be a first step in the comparison of bank based bubbles, as in the 2007 banking crisis with household-based ones, as the dotcom crisis. Their results show that, when banks hold the bubble, the economic cost of the crisis is much higher, in line with the empirical results of Jorda et al. (2015).

Aoki and Nikolov assume, as in the previous models, that firms' future income is only partially pledgeable and consider, in addition, as Calomiris and Kahn (1991) or Gertler and Peter (2011) do, that banks are able to divert a fraction of their assets. Thus, the severity of asymmetric information (and the credit frictions that it implies) is given by the fraction bankers can divert and the fraction of collateral that can be repossessed by creditors in case of default.

Because of firms' limited collateral, there is a shortage of investment opportunities which creates a "savings glut." This may lead to the financing of low productivity firms that are able to post some collateral to the detriment of high productivity firms that, in spite of constant returns to scale, are rationed by their collateral, a result that is reminiscent of the credit traps model of Matsuyama (2007). In such a context, the bubbles role is to provide an additional investment channel in an economy where, otherwise, the excess of savings with respect to the profitable investment opportunities would lead to an inefficient allocation. When this is the case, bubbles increase bank lending and firms benefit from it.

The main contribution of Aoki and Nikolov's model is that the impact of bubbles on the equilibrium allocation depends upon who holds the asset subject to bubbles. When banks are investing in bubbles, there is an output boom that last for as long as the bubble does not burst, but when the bubble bursts, it triggers a deeper recession

In the absence of government support, banks investment opportunities are superior to those available to the savers. Thus, banks prefer to invest in the productive economy, while savers are willing to take the risk of investing in the bubble, as the expected return is larger than the deposit.

The allocation changes completely when a banks' bailout policy by the government is expected. In this case, this provides a subsidized put to banks and increases their incentives to hold the bubble. This is the case because in a bailout, banks will recover a fraction of the investment they made on the bubble. Because the value of the put is higher the higher banks' leverage, banks will hold bubbles when they are highly leveraged, when long-term real interest rates are low and when they have high deposit insurance subsidies. The evolution of banks' balance sheet will then be the one described by Blanchard-Watson and banks will experience strong growth of net worth while the bubble survives and a sharp fall in net worth or bankruptcy when it bursts.

Banks' Liquidity

The above models do not address the issue of the link between credit supply and bubble creation. Whether the interest rate is exogenously given as in Farhi and Tirole or is determined by the marginal productivity of capital or the intertemporal rate of substitution, shocks on the supply of credit are absent so that their effect on both interest rate and the equilibrium price of the bubble cannot be assessed. When financial intermediation is introduced, banks' ability to access funds and, therefore, to supply credit will determine the behavior of households that will decide whether to deposit or borrow to invest in the bubble.

Contrary to the usual assumptions regarding the financial market imperfections, in Freixas and Perez-Reyna (2017), firms have sufficient collateral and, in equilibrium, their marginal profit from investing in bubbles equals the lending interest rate. Still, the no-arbitrage condition imposes the equality between the gross return on buying the bubble, [E.sub.t]([p.sub.t+1]/[p.sub.t]) and either the deposit rate, if households do not borrow to invest in the bubble or the lending rate if the current price of the bubble [p.sub.t] is higher than the household's income. This implies different regimes are possible, which is consistent with different types of crises, as in Aoki and Nikolov, as the leveraged regime could trigger a banking crisis, while there is no banking risk when households buy the bubble out of their income.

This framework may be interesting for several reasons: first it shares some of the characteristics of any real estate boom, where banks do not hold real estate but do lend to households by offering mortgages and face the risk of the collapse of real estate prices. Second, it allows us to consider whether bubbles in household hands may have a positive effect on the efficiency of the resulting allocation. Third, this framework allows to explore some of the characteristics of the optimal macroprudential policy.

Within each regime, the determination of the equilibrium establishes a relationship between today's prices and interest rates and the future expected price for the bubble. So, in equilibrium these two variables will be jointly determined. In the steady-state case, where the expected future price for the bubble is constant, a higher supply of credit will lead to a lower equilibrium interest rate and a higher price for the bubble. Still, if the asset price follows a random walk, the interest rate may be zero. As mentioned, the probability distribution of the exogenous random variables (productivity, liquidity, etc.) that determine the equilibrium will be crucial in the formation of expectations and the equilibrium, a point that is directly related to the design of a macroprudential policy.

Contrary to Farhi and Tirole, bubbles will be larger when liquidity is higher, and contrary to Martin and Ventura, a high realized productivity shock reduces the size of the bubble. The result is not surprising as the role of the bubble is reversed when it is held by entrepreneurs as collateral. To see why the effect is the opposite, consider, for instance, the case of scarce liquidity, in the Farhi and Tirole model, where there will be a demand for bubbles by young entrepreneurs so that, as old, they will have more collateral. If instead bubbles are an alternative way to store value, then their role will be to drain excess collateral and scarce liquidity will lead to higher interest rates so that the price of the bubble will drop. The prediction is, therefore, that bubbles will emerge when interest rates are low and banking liquidity is high, in line with the results of Jorda et al. (2011). In our framework, excess liquidity results in systemic risk, because the bubbles, even if held by households, will be financed by banks' credit.

Macroprudential Policy Implications: Through the Bubble Lens

A normative approach to macroprudential regulation has to consider the cost benefit analysis of banks' risk taking. The theory of rational bubbles constitutes an interesting approach compatible with the existing empirical evidence that allow us to put to test a number of intuitions and gain additional insights.

Because it is possible to define a measure of efficiency and ex ante welfare, the assertion that bubbles crowd out investment is no longer true (Farhi and Tirole 2012; Martin and Ventura 2012) or may be efficient (Freixas and Perez-Reyna 2017). The models briefly outline in the previous section enrich the debate on macroprudential regulation. While the standard view states that there is a cost in implementing a macroprudential policy targeted at reducing the risk of a bubble bursting, in an overlapping generation model with credit frictions this may no longer be true.

More specifically, the rational expectations approach models allow, so far, to draw four important lessons to take into account in the design of macroprudential policy.

First, implementing a macroprudential policy changes the whole equilibrium dynamics, and therefore, the analysis has to consider how the very announcement of a credible macroprudential policy will change the no-arbitrage condition, as the expected value of the bubble will be affected and this, in turn, will change the equilibrium outcome, even if macroprudential policy is only to be exerted conditional on the bubble reaching some threshold.

Second, the lessons on macroprudential policy from Martin and Ventura (2016) result from their finding that there is an optimal level of the bubble. Macroprudential regulation should therefore introduce lump sum taxes and transfers from young entrepreneurs to old entrepreneurs if positive, and the reverse if negative. The policy will be of the "leaning against the wind" type and will consist in taxing credit when collateral is excessive and subsidizing its when collateral is scarce. Consequently, when the bubble is large, macroprudential policy both reduces systemic risk and improves resource allocation.

Third, an efficient macroprudential policy should take into account who holds the bubble, as pointed out by Aoki and Nikolov (2015) and illustrated by the difference between a dotcom bubble and the 2008 crisis.

Finally, as illustrated in Freixas and Perez-Reyna (2017) a macroprudential policy that leads to reducing the risk in interest rates improves productive efficiency. This implies that macroprudential policy has to distinguish and treat differently liquidity shocks and productivity shocks.

Such lessons are to be taken with caution as the economic environment regulators are confronted with is a much more complex one. Still the implication is that additional models are to be considered, whether with multiple bubbly assets, irrational investors, non-stationary economies and the like. At this stage, we believe the road is open for new developments that will affect our view of macroprudential policy in the future.

Conclusion

To sum up, the analysis of overlapping generation bubbles allows us to make explicit the self-reinforcing cycle between credit and the equilibrium price of assets. The approach brings new insights into the relationship between asset prices, credit expansion, banks' risks and systemic risks that match some of the recent empirical evidence on the macroeconomic underpinning of systemic crises.

Different approaches emphasize different potential roles for bubbles, whether as collateral or as a liquidity buffer. The conclusions for macroprudential policy are still tentative, yet the approach constitutes a clearly defined way to model the dynamics of the build up of endogenous systemic risk. Still, in spite of the insights the theory provides, there is a long way to go until it is possible to test the different approaches and see about their possible application in the design of macroprudential policy.

https://doi.org/10.1057/s41294-018-0054-8

Published online: 13 February 2018

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(1) Barcelona, Spain

(1) In Laeven and Valencia (2012), a systemic financial crisis is defined as an event during which a country's banking sector experiences bank runs, sharp increases in default rates accompanied by large losses of capital that result in public intervention, bankruptcy or the forced merger of major financial institutions. Regarding credit booms. Dell'Ariccia et al. (2012) define it either as a deviation from trend greater than 1.5 times the standard deviation and an annual growth rate of the credit to GDP ratio that exceeds 10% or an annual growth rate of the credit to GDP ratio exceeding 20%.

(2) In addition, other conditions may impose a cap on the bubble price. Thus, for instance, the existence of substitutes for the bubble may make it impossible for an explosive bubble to exist or may imply that the substitutes prices follow the same pattern.

(3) Abel et al. (1989) test whether this condition holds in developed economies and show that the productive sector disgorges at least as much as it invests. Their conclusion that the economy is dynamically efficient is, nevertheless, only valid provided financial markets are efficient, a very restrictive assumption.

(4) Contrary to standard liquidity models, because of the multiplicative effect of liquidity on investment, the steady-state interest rate increases with the outside liquidity.

Xavier Freixas (1)

[mail] Xavier Freixas

xavier.freixas@upf.edu
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