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Notes on Bubbles J. Bradford DeLong University of California at Berkeley and NBER brad.delong@gmail.com http://delong.typepad.com/ +1 925 708 0467 April 21, 2009 Let us begin with a very long quote from Chrles Kindleberger, who in turn begins with Hyman Minsky: We start with the model of the late Hyman Minsky, a man with a reputation among monetary theorists for being particularly pessimistic, even lugubrious, in his emphasis on the fragility of the monetary system and its propensity to disaster. Although Minsky was a monetary theorist rather than an economic historian, his model lends itself effectively to the interpretation of economic and financial history. Indeed, in its emphasis on the instability of the credit system, it is a lineal descendant of a model, set out with personal variations, by a host of classical economists including John Stuart Mill, Alfred Marshall, Knut Wicksell, and Irving Fisher. Like Fisher, Minsky attached great importance to the role of debt structures in causing financial difficulties, and especially debt contracted to leverage the acquisition of speculative assets for subsequent resale. According to Minsky, events leading up to a crisis start with a "displacement," some exogenous, outside shock to the macroeconomic system. The nature of this displacement

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varies… the outbreak or end of a war, a bumper harvest or crop failure, the widespread adoption of an invention with pervasive effects---canals, railroads, the automobile---some political event or surprising financial success… an unanticipated change of monetary policy…. Displacement brings opportunities for profit in some new or existing lines and closes out others. As a result, business firms and individuals with savings or credit seek to take advantage of the former and retreat from the latter. If the new opportunities dominate those that lose, investment and production pick up. A boom is under way. In Minsky’s model, the boom is fed by an expansion of bank credit that enlarges the total money supply…. For a given banking system at a given time, monetary means of payment may be expanded not only within the existing system of banks but also by the formation of new banks, the development of new credit instruments, and the expansion of personal credit outside of banks. Crucial questions of policy turn on how to control all these avenues of monetary expansion. But even if the instability of old and potential new banks were corrected, instability of personal credit would remain to provide means of payment to finance the boom, given a sufficiently throughgoing stimulus. Let us assume, then, that the urge to speculate is present and transmuted into effective demand for goods or financial assets…. Prices increase, giving rise to new profit opportunities and attracting still further firms and investors. Positive feedback develops, as new investment leads to increases in income that stimulate further investment and further income increases. At this stage we may well get what Minsky called "euphoria." Speculation for price increases is added to investment for production and sale. If this process builds up, the result is often, though not

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inevitably, what Adam Smith and his contemporaries called "overtrading." Now, overtrading is by no means a clear concept. It may involve pure speculation for a price rise, an overestimate of prospective returns, or excessive "gearing."... Overestimation of profits comes from euphoria, affects firms engaged in the production and distributive processes, and requires no explanation. Excessive gearing arises from cash requirements that are low relative both to the prevailing price of a good or asset and to possible changes in its price. It means buying on margin, or by installments, under circumstances in which one can sell the asset and transfer with it the obligation to make future payments. As firms or households see others making profits from speculative purchases and resales, they tend to follow: "Monkey see, monkey do." In my talks about financial crisis over the last decades, I have polished one line that always gets a nervous laugh: "There is nothing so disturbing to one’s well-being and judgment as to see a friend get rich.” When the number of firms and households indulging in these practices grows large, bringing in segments of the population that are normally aloof from such ventures, speculation for profit leads away from normal, rational behavior to what has been described as "manias" or "bubbles." The word mania emphasizes the irrationality; bubble foreshadows the bursting…. At a late stage, speculation tends to detach itself from really valuable objects and turn to delusive ones. A larger and larger group of people seeks to become rich without a real understanding of the processes involved. Not surprisingly, swindlers and catchpenny schemes flourish…. As the speculative boom continues, interest rates, velocity of circulation, and prices all continue to mount. At some

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stage, a few insiders decide to take their profits and sell out. At the top of the market there is hesitation, as new recruits to speculation are balanced by insiders who withdraw. Prices begin to level off. There may then ensue an uneasy period of "financial distress." The term comes from corporate finance, where a firm is said to be in financial distress when it must contemplate the possibility, perhaps only a remote one, that it will not be able to meet its liabilities. For an economy as a whole, the equivalent is the awareness on the part of a considerable segment of the speculating community that a rush for liquidity---to get out of other assets and into money---may develop, with disastrous consequences for the prices of goods and securities, and leaving some speculative borrowers unable to pay off their loans‌. The specific signal that precipitates the crisis may be the failure of a bank or firm stretched too tight, the revelation of a swindle or defalcation by someone who sought to escape distress by dishonest means, or a fall in the price of the primary object of speculation as it, at first alone, is seen to be overpriced. In any case, the rush is on. Prices decline. Bankruptcies increase. Liquidation sometimes is orderly but may degenerate into panic as the realization spreads that there is only so much money, not enough to enable everyone to sell out at the top. The word for this state--again, not from Minsky---is revulsion. Revulsion against commodities or securities leads banks to cease lending on the collateral of such assets. In the early nineteenth century this condition was known as discredit. Overtrading, revulsion, discredit—all these terms have a musty, oldfashion flavor. They are imprecise, but they do convey a graphic picture. Revulsion and discredit may go so far as to lead to panic (or as the Germans put it, Torschlusspanik. "door-shut-

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panic"), with people crowding to get through the door before it slams shut. The panic feeds on itself, as did the speculation, until one or more of three things happen: (1) prices fall so low that people are again tempted to move back into less liquid assets; (2) trade is cut off by setting limits on price declines, shutting down exchanges, or otherwise closing trading; or (3) a lender of last resort succeeds in convincing the market that money will be made available in sufficient volume to meet the demand for cash. Confidence may be restored even if a large volume of money is not issued against other assets; the mere knowledge that one can get money is frequently sufficient to moderate or eliminate the desire. Whether there should be a lender of last resort is a matter of some debate. Those who oppose the function argue that it encourages speculation in the first place. Supporters worry more about the current crisis than about forestalling some future one... This is, I think, right. And here I have a problem. For it is pretty clear to me that the conventional model of bubbles is not terribly illuminating as a model of this process. The conventional model of bubbles starts with the assumption of a constant required rate of return r. It continues with a one-period equilibrium condition for the price of an asset paying a dividend dt. If there is even one rational, utility-maximizing agent in the economy, than for that agent: (1)

(1 + r)( p t − d t ) = E t p t +1

Solving:

â‚Ź

(2)

p t = dt +

E t p t +1 1+ r

â‚Ź

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Iterating forward, assuming that there is at least one rational utilitymaximizing agent buying and selling at each period, identifying “expectation” with orthogonal projection onto some information set, and using the law of iterated projections:

(3)

  E p d d p t = E t d t + t +1 + t+2 2 + ... + t t+nn 1 + r (1 + r)   (1 + r)

(4)

  n    E p  d t +i    p t = limE t ∑  + lim   t t+ni  i  n−>∞ n−>∞   i=0  (1 + r)   (1 + r) 

The first of these terms is the fundamental. The last is the bubble term. If we call the rate of growth of the economy g and say that prices in the far future€are some stationary variable η blown up by the size of the economy: (5)

p t+n = ηt+n (1 + g) n

Then (4) becomes:

€ (6)

n   n  (1 + g)  p t = limE t ∑ d t+i  + limE t ηt+n  n n−>∞  n−>∞ (1 + r)    i=0

It is plain that if g is greater than r bubbles are not just “rational” and possible—they are required. If g is equal to r—or, rather, if r is equal to g, € are forces in the economy that drive the two together—than if there bubbles are once again required unless there are forces in the economy that drive lim(Et(ηt+n) = 0 as well. But if r > g than prices must be equal to fundamentals—if the rule by which we identify “expectation” with orthogonal projection and the law of iterated orthogonal projections holds. So we must find some way to drive r less than g—which could be done with a dynamically-inefficient economy, which requires something like an

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overlapping-generations framework to block the long-horizon investments and trades between the present and the far future that would eliminate dynamic inefficiency; or could be done by depriving “rational� investors of capital so that when asset prices become high (or low) relative to fundamentals, their attempts to profit from this by taking on leveraged positions very quickly drive their r to large and negative (or large and positive) values. So let us take two alternative roads to understanding the asset price fluctuations we are now seeing.

The first road is indeed to focus on limits to arbitrage and to deprive our rational agents of the power to make prices right. The idea is that of in DeLong et al. (1990). Our basic model is a stripped down overlapping generations model with two-period lived agents (Samuelson 1958). For simplicity, there is no first period consumption, no labor supply decision, and no bequest. As a result, the resources agents have to invest are exogenous. The only decision agents make is to choose a portfolio when young. The economy contains two assets that pay identical dividends. One

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of the assets, the safe asset (s)—government bonds—pays a fixed real dividend r. Asset (s) is in perfectly elastic supply: a unit of it can be created out of and a unit of it turned back into a unit of the consumption good in any period. Taking consumption each period as numeraire, the price of the safe asset is always fixed at one. The dividend r paid on asset (s) is thus the riskless rate. The other asset, the unsafe asset (u)—aggregate equities—always pays the same fixed real dividend r as asset (s). But (u) is not in elastic supply: it is in fixed and unchangeable quantity, normalized at one unit, so the price of (u) is a variable p. If the price of each asset were equal to the net present value of its future dividends, then assets (u) and (s) would be perfect substitutes and would sell for the same price of one in all periods. But this is not how the price of (u) is determined in the presence of noise traders. There are two types of agents: sophisticated investors (denoted “i”) who have rational expectations and noise traders (denoted “n”). We assume that noise traders are present in the model in measure µ, that sophisticated investors are present in measure 1-µ, and that all agents of a given type are identical. Both types of agents choose their portfolios when young to maximize perceived expected utility given their own beliefs about the exante mean of the distribution of the price of (u) at t+1. The representative sophisticated investor young in period t accurately perceives the distribution of returns from holding the risky asset, and so maximizes expected utility given that distribution. The representative noise trader young in period t misperceives the expected price of the risky asset by an i.i.d. normal random variable ρ: We normally distributed shocks and constant absolute risk aversion, oneperiod ahead utility maximization is the same as maximizing a meanvariance utility function: (7) where the only risk is uncertainty in period t about the price in period t+1:

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(8) Demands for the risky asset are then:

(9) And prices are, given supply fixed at one unit of the risky asset: (10) The only variable term is the second, and so we can solve through for the variance:

(11) The first term, of course, is the fundamental value. The other three are “bubble” terms: deviations of prices in fundamentals in some sense (although note that equation (1) still does hold for the rational agents for their own individual marginal required rates of return). The second term captures the fluctuations in the price of the risky asset (u) due to the variation of noise traders’ misperceptions. Even though asset (u) is not subject to any fundamental uncertainty and is so known by a large class of investors, its price varies substantially as noise traders’ opinions shift. When a generation of noise traders is more “bullish” than the average generation, they bid up the price of (u). When they are more bearish than average, they bid down the price. The third term captures the deviations of p from its fundamental value due to the fact that the average

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misperception by noise traders is not zero. If noise traders are bullish on average, this “price pressure” effect makes the price of the risky asset higher than it would otherwise be as irrationally exuberant noise traders bear a greater than average share of price risk and hence lower the required rate of return for rational, sophisticated investors. The final term is the heart of the model. Sophisticated investors would not hold the risky asset unless compensated for bearing the risk that noise traders will become bearish and the price of the risky asset will fall. Both noise traders and sophisticated investors present in period t that asset (u) is mispriced, but because pt+1 is uncertain neither group is willing to bet too much on this mispricing. At the margin, the return from enlarging one’s position in an asset that everyone agrees is mispriced (but different types think is mispriced in different directions) is offset by the additional price risk that must be run. Noise traders thus “create their own space”: the uncertainty over what next period’s noise traders will believe makes the otherwise riskless asset (u) risky, and drives its price down and its return up. This is so despite the fact that both sophisticated investors and noise traders always hold portfolios which possess the same amount of fundamental risk: zero. Any intuition to the effect that investors in the risky asset “ought” to receive higher expected returns because they perform the valuable social function of risk bearing neglects to consider that noise traders’ speculation is the only source of risk. For the economy as a whole, there is no risk to be borne. To make this magic trick work—thus pulling of variance out of fixed fundamentals—we need all three of: • • •

Short horizons Infinite future extent Unbounded returns—in particular, unbounded negative prices.

Friedman (1953) argued that noise traders who affect prices earn lower returns than the sophisticated investors they trade with, and so economic selection works to weed them out. But here it need not be the case that noise traders earn lower returns. Noise traders’ collective shifts of opinion

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increase the riskiness of returns to assets. If noise traders’ portfolios are concentrated in assets subject to noise trader risk, noise traders can earn a higher average rate of return on their portfolios than sophisticated investors. Relative expected returns are:

(12) Note that if the noise trader share is zero and if noise traders are optimistic on equities on average, their returns will always be higher in expectation than rational investors. There is a hole in the argument of Friedman (1953): the rational portfolio produces higher returns on average, but it does not necessarily produce higher utility. This paper is written in the rhetorical mode of the magician: that of pulling a rabbit out of a hot—a “nothing up our sleeve” rhetoric. Does it apply? A good question. I would argue that yes it does. But this just tells us that we can build bubbles if we can impose short horizons and limited capital on sophisticated investors even if, in some sense, the safe r is greater than the economy’s growth rate g. But how do these asset bubbles behave? Why do they cause trouble? Second, let’s take a different tack. Start with a unit interval [0, 1] of investors. Investors can either be in the market and own stocks or out of the markets and own bonds. Bonds pay a fixed and certain net rate of return r each period. Stocks have a price p, and each period pay a serially-uncorrelated stochastic dividend d: δ with probability π and 0 with probability 1-π. When investors want to buy stocks they are supplied by venture capitalists. When investors want to sell stocks they are bought up and destroyed by private equity firms. By coincidence the price at which venture capitalists and private equity firms create and destroy equities is such that the price of equities p is equal to the fraction p of agents who own stocks.

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At the start of each period t a value for the dividend and a price pt for stocks is cried out by a Walrasian auctioneer. A parameter λ governs the fraction of agents are paying attention: those who are paying attention frantically calculate the rate of return on stocks (pt - pt-1 + d)/pt-1. Based on whether that return was more or less than the rate of return on bonds some of the λ agents who are paying attention commit to buying stocks or selling stocks, the fraction p of stockholders shifts, and at the start of the next period the auctioneer will cry out a new value for p:

(13)

  p − p t−1 + δ  − r , with probability π  p t + λp t (1− p t ) t  p t−1   p t+1 =     p + λp (1− p ) p t − p t −1 − r , with probability 1-π  t t   t  p t−1 

The intuition is that (a) you have to encounter somebody following a different € portfolio strategy than you to even think of switching, and (b) once you think of switching you do so depending on whether stocks or bonds did better recently. Taking expectations: (14)

 p − p t−1 + πδ  E t ( p t +1 ) = p t + λp t (1− p t ) t − r p t−1  

(15)

E(Δp t +1 ) = 0 if p t−1 =

So: €

πδ r

is the “fundamental” value of p: call it p*. €

Note that we can eliminate the possibility of bubbles in this model by having investors ignore the capital gain when they assess returns, as in:

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(16)

    p t + λp t (1− p t ) δ − r , with probability π p t+1 =   p t−1   p + λp (1− p )(−r), with probability 1-π  t t t

Suppose first that we have no uncertainty: π=1. But also suppose that something has just happened—a global savings glut, perhaps—to drop r € from its initial level of 12.5% (corresponding to a p* of 0.2) to a level of 5% (corresponding to a p* of 0.5). For a value of λ = 1.5 we get overshoot and then slow decline to the fundamental value with the behavioral equation (13); we get slow convergence with the behavioral equation (16).

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Stock Market Prices 0.9 0.8 0.7

Value

0.6 0.5 0.4 0.3 0.2 0.1 0 0

20

40

60

80

100

Period

Back when my great uncle Phil briefly had me in the investment management business, I had to talk to accountants—accountants who strongly believed in “book value.” Every time I asked them to prepare a set of market value accounts, at the bottom of every page there would be: “THIS REPORT WAS NOT PREPARED ACCORDING TO GENERALLY-ACCEPTED ACCOUNTING PRINCIPLES.” We get a mania—but not a panic or a crisis. We get a bubble—but it does not pop, instead it deflates.

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Stock Market Prices 0.9 0.8 0.7

Value

0.6 0.5 0.4 0.3 0.2 0.1 0 0

20

40

60

80

100

Period

Now let’s add back in some period-to-period noise. Let’s halve the probability that the dividend is actually paid to 1/2 but double the amount of the dividend to .05—so that we still have a fundamental value p* of 0.5. This initial “displacement”—the fall in r from 12.5% to 5%--is a large impetus. For these parameter values it induces rapid convergence, substantial overreaction, and then once again deflation back toward fundamentals, which are then reached about period 30. But asset prices then do not then stay there.

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Note the behavior of this simulation after period 30, when the deterministic system has settled down to within a small neighborhood of its fundamental value p*. The fact that porfolio choice depends not just on fundamental returns but on capital gains—themselves driven by previous fundamental displacements and previous capital gains—amplifies the fluctuations in prices considerably. The lesson is that if you have short look-back periods, poor knowledge of long-run fundamentals, a tendency to adopt recently-successful strategies, moderately long-duration assets traded at some frequency, and an initial displacement—a true change in long-run fundamentals—it is easy to generate bubbles and manias and overshooting on the upside and on the downside. If we take a look at our equation (13) transformed into lag- and difference-operator form: (17)

Dp t+1 / p t = λ (1− p t )L(Dp t+1 / p t ) + λ (1− p t )(X − r)

and transform it: €

(18)

D 2 p t+1 / p t = [ λ (1− p t ) − 1] L(Dp t+1 / p t ) + λ(1− p t )(X − r)

We see that when λ=0 we get smooth exponential convergence. We see that when € λ(1-p)=1, we have a harmonic oscillator: we then get undamped oscillations that return the price to fundamentals relatively quickly, but then shoot on past and come back again for another try. In either case the market does not seem to be performing particularly well as a social capital allocation and forecasting mechanism.

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Stock Market Prices

Stock Market Prices

1.2

1 0.9

1

0.8 0.7

0.8

Value

Value

0.6

0.6

0.5 0.4

0.4

0.3 0.2

0.2

0.1

0

0

0

20

40

60

80

100

0

20

40

Period

60

80

100

80

100

Period

Stock Market Prices

Stock Market Prices

0.8

1 0.9

0.7

0.8 0.6 0.7 0.6 Value

Value

0.5 0.4

0.5 0.4

0.3

0.3 0.2 0.2 0.1

0.1

0

0 0

20

40

60

80

100

0

Period

20

40

60 Period

One last remark: we have manias, bubbles, and overshootings. But that is not what we really want out of a model. Our model is not sufficient to generate panics and crashes. Panics and crashes are asymmetrically strong downward movements. We cannot generate them out of a model that is, after all, a symmetric one. The place to go, I think, to get panics and crashes—sharper and steeper and larger downward than upward movements—is to note the asymmetry

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between profits and losses. Profits make you feel exuberant—rationally or irrationally—and prone to expand your position. Losses leave you bankrupt and your position is then involuntarily sold out from under you. Portfolio insurance, stop-loss orders, margin calls, capital requirements, and many other contractual mechanisms work to make positive-feedback trading on the downside automatic and hence swift, while positivefeedback trading on the upside remains discretionary. How to get a financial market to do better? Patient, long-run capital. But patient, long-run capital is limited by (a) agency, (b) desire to reveal and exhibit skill, (c) limits to arbitrage, (d) leverage, and (e) noise trader risk.

Franklin Allen and Douglas Gale (2000), "Bubbles and Crises," Economic Journal 110:460 (Jan.) http://www.jstor.org/stable/pdfplus/2565656.pdf Michael Bordo and Olivier Jeanne (2002), "Monetary Policy And Asset Prices: Does 'Benign Neglect' Make Sense?," International Finance http://www.nber.org/papers/w8966.pdf

Charles Kindleberger, Manias, Panics and Crashes J. Bradford DeLong, Andrei Shleifer, Lawrence Summers, and Robert Waldmann (1990), “Noise Trader Risk in Financial Markets.”

Morris Goldstein (1998), "The Asian Financial Crisis: Causes, Cures and Systemic Implications" (Online reading only; printing not permitted) http://bookstore.petersoninstitute.org/book-store/22.html

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Paul Krugman (1999), "Analytical Afterthoughts on the Asian Crisis" http://www.sfu.ca/~kkasa/krugman1.pdf Paul Krugman (1998), "What Happened to Asia?" http://web.mit.edu/krugman/www/DISINTER.html

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DeLong, "Notes on Bubbles" (April 21, 2009)  

Let us begin with a very long quote from Chrles Kindleberger, who in turn begins with Hyman Minsky: J. Bradford DeLong According to Minsky,...

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