1. business and industrial

The Impact of Capital Gain Taxes and Transaction Costs on
Asset Bubbles
Yue Shen∗†
Queen’s University
May 29, 2015
Abstract
In this paper we evaluate the effects of taxes and transaction costs on asset bubbles. We construct
a model of asset bubbles by incorporating purchase into the framework of Abreu and Brunnermeier
(2003) so that capital gains can be evaluated. We find that the capital gain tax has no effect on
the size of the bubble when there is perfect capital loss tax credit, and the bubble size is decreasing
in the capital gain tax when there is no tax credit. Therefore dealing with bubbles with the capital
gain tax not only requires imposing the tax, but also requires tightening the policies on tax credit.
In addition, the size of the bubble is decreasing in the transaction cost and the return from outside
option. This implies that central banks’ low interest policies could induce bubbles in, for instance,
real estate and stock markets. To show that our model can potentially be tested, we explore several
historical bubbles and identify their actual sizes from data. After normalization, we compare these
actual sizes of bubbles and fit them into theoretical results derived from the model.
1
Introduction
Since the 2008 US subprime mortgage crisis, housing prices in certain major cities outside the United States
have been steadily increasing at speeds higher than their historical norms, including those in London (UK),
Vancouver, Toronto, Beijing, Shanghai, Hong Kong1 , and most capital cities in Australia. For example,
house prices in London rose by 18% in 2013 alone. The upsurging prices have raised serious concerns that
bubbles are developing in these cities. Although the ongoing discussion on macroprudential policies might
help regulate domestic financial practitioners, these policies seem to be ineffective when international
“hot money” and private investors contribute greatly to the bubbles, and there is no consensus on the
implementation of these policy tools. A different choice would be a tax. There is a large literature on the
∗
I am grateful to James Bergin and Frank Milne for numerous valuable comments. I thank Ruqu Wang, Marco Cozzi,
Rui Gao, Steve Kivinen, Marie-Louise Vierø, Tianyi Wang, Jan Zabojnik, Chenggang Zhou, and participants in CEA and
department seminars at Queen’s. All errors are my own.
†
Department of Economics, Queen’s University, Kingston, Ontario, Canada. [email protected]
1
Housing prices in hundreds of smaller cities in China are also inflating rapidly.
effects of financial transaction taxes on price volatilities, but to our knowledge the effectiveness of taxes
on asset bubbles, especially capital gains tax, has not been studied by the existing literature.
With or without economic theories, governments and law makers are ready to intervene, or have already
done so. Chancellor George Osborne of the British parliament said that as of April 2015 he would introduce
a capital gains tax on future gains made by non-residents who sell residential properties in the United
Kingdom2 . In 2013 the Chinese government introduced a 20% tax on capital gains from selling residential
properties if a family owns multiple properties. There is an urgent need to assess the effectiveness of these
tax policies on reducing bubbles.
We evaluate the effects of capital gains tax and transaction costs on asset bubbles. Our model incorporates purchases into the framework of Abreu and Brunnermeier (2003) (henceforth AB2003) and transforms
the uncertainty from the dimension of time in AB2003 to the dimension of value/price. The bubble arises
from the interaction between rational traders and behavioral agents. The asset fundamental jumps up to
an unobservable value, which can be interpreted as a positive productivity shock. Rational traders have
private belief about the new fundamental and their purchases continuously drive up the price. When the
price rises above the underlying fundamental value, a bubble emerges. But since a rational trader does
not know how many others have higher (or lower) beliefs than her, there is no common knowledge about
the emergence of the bubble, which is the key ingredient that induces a bubble. When all rational traders
believe that the price is too high and have stopped buying, the bubble bursts.
The framework of AB2003 is shown to be consistent with recent empirical studies of stock market
data. Temin and Voth (2004) show that a major investor in the South Sea bubble knew that a bubble
was in progress and nonetheless invested in the stock and hence was profiting from riding the bubble.
Brunnermeier and Nagel (2004) and Griffin et al. (2011) both study the tech bubble in the late 1990s.
They show that instead of correcting the price bubble, hedge funds turned out to be the most aggressive
investors. They profited during the upturn and unloaded their positions before the downturn.
Our model features the following dynamics in equilibrium. At the beginning all rational traders try
to buy the asset from a passive behavioral agent. As the price rises continuously, traders with the lowest
beliefs stop purchasing and simply hold while others are buying. As the price is further driven up, lowbelief traders start to sell, intermediate-belief traders stop buying and simply hold, and only high-belief
traders are still buying. The bubble bursts when the highest type stops buying. Rational traders can buy
and sell at any time, but a bubble exists in the unique equilibrium and traders ride the bubble and try to
2
See Chancellor George Osborne’s Autumn Statement 2013 speech on 5 December 2013 at the UK Parliament.
2
sell before the crash to make a profit. Each trader uses a trigger strategy that consists of a threshold for
selling and a threshold to stop buying. A trader holds the asset (if she has bought the asset) and waits,
and when the price reaches her optimal selling threshold she sells all shares. The optimal selling threshold
trades off the marginal benefit of waiting a bit longer to capture more price appreciation with the marginal
cost from the risk of being caught in the crash. Knowing her selling plan, when the price reaches her
stop-buy threshold, a trader stops buying (if she has not been able to buy due to the exogenous market
friction).
In the model rational traders first buy the asset and then sell, as a result of which we can calculate
the net gains and evaluate the effects of capital gains tax. When imposing capital gains tax, we consider
two cases: with perfect capital loss tax credit and without any tax credit3 . In our model, the capital loss
tax credit is perfect if there is no dollar ceiling on the credit and the credit can be carried backward and
forward indefinitely to offset capital gains4 . In this case, the tax benefit (tax exempted due to the credit)
can be regarded as a profit5 . In the unique equilibrium, a bubble exists when the capital gains tax and
fixed transaction cost are not too large.
The first finding of our paper is that capital gains tax has no effect on the size of the bubble when
there is a perfect capital loss tax credit, although the bubble size decreases in the tax rate when there is
no tax credit. The reason is as follows. Consider first the case without tax credit. Recall that the selling
strategies trade off the marginal benefit of waiting with the marginal cost of being caught. When the tax
is imposed, traders face the same potential losses but smaller profits, which advises them to behave more
cautiously and conservatively. In particular, when tax and transaction cost are small, the tax distorts a
rational trader’s selling decision downwards, which in turn squeezes the stop-buy decisions downwards.
Hence the bubble becomes smaller. This can be viewed as an indirect effect of the tax on the bubble.
When the tax and transaction cost are relatively large, the tax does not distort the selling strategy, but a
trader again has to stop buying early because the profit is eroded by the tax and she needs to maintain a
3
Capital loss tax credit is also called capital loss deduction or tax carryover. Due to the risky nature of investment,
fluctuations in returns and profits are almost unavoidable in almost all markets, and most tax systems have a capital loss
tax credit to help smooth these fluctuations. When an investor incurs a capital loss, she has to apply it against other capital
gains for that year. If she still has a loss, then she is entitled to a certain credit to carry forward or backward across tax
years. If she has a taxable capital gain next year, the credit can offset the gain on a dollar-to-dollar basis so that only the
net gain (if any) is subject to tax.
4
In the United States, an individual who incurs capital losses can use the credit to offset any taxable income up to $3000
each year, and carry the unused credit forward indefinitely. A corporation can only offset capital gains (not other profit) with
capital losses and carry it 3 years backward and 5 years forward, without a dollar limit. In Canada the tax code is similar,
except that there is no $3000 limit on individuals, but only 50% of the capital gains/losses are included in the tax base for
both individuals and corporations, and the tax credit can be carried 3 years backward and forward indefinitely.
5
Constantinides (1983) treats this tax benefit as a profit.
3
minimal expected payoff due to an outside option. This is the direct effect of tax on the bubble, which also
deflates the bubble. When there is a perfect tax credit, the tax credit provides a compensation for the loss
such that the loss is also “taxed” and becomes smaller. In this case, a perfect tax credit eliminates both
the direct and indirect effects of the capital gains tax on the bubble, and the bubble becomes immune to
the tax. Thus our paper suggests that to fight bubbles with capital gains taxes, tax authorities should also
tighten their policies on tax credits.
The second finding is that the transaction cost and an outside option (returns from alternative investment opportunities) can also depress the bubble. The cost and the outside option do not affect selling
decisions, but a small cost or outside option has very large marginal depressing effect on stop-buy decisions.
This result backs the practice in some countries to reserve a financial transaction tax (though at very low
rates). Conversely, lowering the cost or the outside option when they are already small has a significant
inflating effect on the bubble. This rationalizes the observations that stock markets and housing prices
are inflated when central banks set very low interest rates (the outside option), and warns that further
lowering interest rates (to fight shocks from the oil price changes, for instance) could induce even larger
bubbles in these markets.
Our model generates empirically testable predictions. To show how we can test the two main findings,
we explore several historical bubbles. We identify the actual sizes of these bubbles as well as parameters
in our model. To compare the actual actual bubble sizes we normalize by dividing them by the identified
belief dispersion, which is an important factor that determines bubble size. From the identified parameters
we can also calculate the theoretical bubble sizes and compare them to the actual ones. The results so far
are inconclusive since currently the sample size is too small.
Some of the implications are consistent with existing empirical evidences. When there is no tax credit,
an increase in capital gains tax not only reduces demand from rational traders, but sometimes delays their
selling and reduces supply (and hence decreases the trading volume). These effects are called “capitalization
effect” and “lock-in effect”, respectively, in the literature and are documented in empirical studies of tax
incidences such as the Revenue Act of 1978, the Tax Reform Act of 1986 and the Taxpayer Relief Act of
1997 in the United States. We will explore empirical evidences and discuss these implications in detail in
Section 7.
Our model is related to a large literature which studies the effects of taxes on asset prices. Constantinides (1983) shows that with capital gains tax and tax credit investors have incentives to sell assets with
losses immediately and secure tax credits while deferring the selling of assets with gains to put off tax
4
payment. His research focuses on trading strategies under stochastic shocks whereas we focus on strategies
facing asset bubbles and crashes with private beliefs. In terms of the effects of the transaction tax on the
volatility of asset prices, Westerhoff and Dieci (2006) show that the transaction tax can stabilize prices,
whereas others suggest that the tax actually amplifies volatility (e.g. Lanne and Vesala (2010)). Empirical
evidences show that ability of the tax to reduce volatility is very limited (e.g. Umlauf (1993) and Hu
(1998)).
On the other hand, research on the effects of taxes on bubbles is scarce. Scheinkman and Xiong (2003)
show that the financial transaction tax can substantially reduce speculative trading volume, but has only
a limited impact on the size of the bubble. Our paper explicitly models an asset bubble and evaluates the
effects of capital gains tax on trading behavior and the size of bubble, and is complementary to the above
literature in understanding the effects of taxes on financial markets.
From a modeling perspective, our paper is related to a quickly growing literature on bubbles. For
surveys, refer to Brunnermeier (2008), Brunnermeier (2001), Brunnermeier and Oehmke (2012) and Xiong
(2013). AB2003 relies on the asynchronous timing of awareness to generate bubbles, whereas we transform
the uncertainty from time to value/price and remove their assumption of sequential awareness. AB2003
requires an exogenous exponential price path that grows at an accelerating rate, whereas our model allows
for any continuous and strictly increasing price path and is therefore more robust in this sense. In AB2003
the price rises due to behavioral agents. Strategic traders do not raise the price directly but only “allow”
the bubble to grow. Our model incorporates purchases into their framework such that the price keeps
increasing exactly because rational traders are buying. The burst is triggered when no one wants to buy
in our model, which is in contrast to AB2003 where an exogenous selling threshold is imposed.
Doblas-Madrid (2012) removes behavioral agents from the framework of AB2003 and instead uses
idiosyncratic liquidity shocks to force rational traders to sell and generates trades when the price is low
and no one wants to sell. The price is determined in equilibrium every period. However, to support an
increasing price path, endowments must also increase over time. Observing the noisy price, agents update
their belief, and when the noise in the price can no longer conceal the downward selling pressure, the bubble
bursts. The model with the price determined by a spot equilibrium every period is definitely realistic, but
is difficult to apply to our setting because we do not have any noise in the price6 .
The remainder of this paper is organized as follows. In Section 2, we introduce the model. Section 3
6
To conceal the very first strategic sale in our model, one must assume that there is an extra amount of money injected
to balance the strategic sale at that moment, which can be difficult to justify.
5
shows how a trader’s strategy space can be reduced so that we have a simple game to solve. In Section 4
we solve an individual trader’s problem backwards, by first solving for a trader’s optimal selling price and
then her stop-buy price. The equilibria with perfect tax credit and without tax credit are characterized in
Sections 5 and 6, respectively. Section 7 provides empirical predictions, studies several historical bubbles
and shows how we can potentially test the model. Section 8 discusses policy implications. Section 9
concludes.
2
The model
Time t is continuous. There are two types of agents: risk neutral rational traders (henceforth traders)
and a large passive (behavioral) agent, though the latter is not our primary interest. There is one asset
(henceforth the asset) and rational traders also have an outside investment option. The asset has a total
supply Q and all shares are held by the passive agent at the beginning. At t = 0 the asset’s fundamental
value jumps up from p0 to θ and does not change henceforth. θ is uniformly distributed on [0, ∞)7 and
is unobservable to both traders or the passive agent. Without loss of generality, we assume p0 = 0. The
outside option provides an expected profit R ≥ 0 which is common knowledge and uncorrelated with θ. A
trader cannot hold the asset and the outside option at the same time, and cannot switch to the other if
she has bought one. Capital gains from both the asset and the outside option are subject to tax, and a
trader will choose to hold the asset only when its expected profit is strictly higher than the outside option.
Without loss of generality each trader’s asset position is restricted and normalized to [0, 1]8 .
The inverse asset supply function of the passive agent is
p(t) = α(Dr (t))
(1)
where p(t) is the price at t which is publicly observable, α(·) is a continuous, strictly increasing function
and Dr (t) is the total shares held by all traders (the aggregate position of all traders) at t. Since the initial
price p(0) = 0, when the fundamental value jumps above 0, traders keep buying the asset from the passive
agent which drives up the price continuously. The passive agent’s inverse supply function implies that, for
every share she sells to traders, she must be compensated at a price higher than the previous share she
sold. This behavior can be interpreted as portfolio diversification requirements or institutional frictions
that make the risky asset more valuable to the passive agent when she holds less and less of the asset.
7
The improper uniform distribution on [0, ∞) has well defined posterior belief when we specify how signals are distributed.
The uniform prior distribution gives tractable solutions and is adapted from Li and Milne (2014).
8
Limited short selling of the asset is allowed for traders.
6
Alternatively it can be interpreted as an adverse selection problem. When traders have private information
about the asset fundamental value and keep buying the asset, it is natural for the uninformed passive agent
to respond by raising the price when selling it, as in Kyle (1985). Similar behavioral asset supplies have
also been adopted by De Long et al. (1990), where passive investors supply the asset at an increasing price
when rational speculators are buying, and by Brunnermeier and Pedersen (2005) and Carlin et al. (2007),
where long-term traders buy the asset when strategic traders’ liquidation pushes down the price and sell
the asset when strategic traders’ buy-back pushes up the price.
In addition, we assume that the price can be arbitrarily high when traders’ aggregate position approaches Q, i.e. lim α(x) = ∞. This guarantees that the price will not stop at a level lower than θ when
x−→Q
traders have bought all shares (recall that θ can be arbitrarily large).
Each trader receives a private signal v which is uniformly distributed on [θ − η2 , θ + η2 ]. There is enough
mass of traders (potentially infinite) for each type v, which is to guarantee that we will not run out of
buyers while the price is still below θ 9 . Let Φ(θ|v) denote the cumulative distribution function of the
belief of a trader v conditioned on v, and φ(θ|v) the corresponding probability density function. Given
a signal v, the posterior belief of a trader is that θ is uniformly distributed on [v − η2 , v + η2 ]. Therefore
φ(θ|v) =
1
η
and Φ(θ|v) =
θ−(v− η2 ) 10
.
η
Figure 1 depicts the posterior belief about θ for trader v, v 0 and v 00 in
the case of uniform prior belief. These different posterior beliefs reflect different opinions about the asset
Legend:
φ(θ|s)
φ(θ|v 0 )
φ(θ|v 00 )
v
v0
s00
φ(θ|v)
φ(θ|v 0 )
φ(θ|v 00 )
θ
Figure 1: Posterior beliefs
fundamental value. A trader is not sure about her signal’s position within [θ − η2 , θ + η2 ], i.e., a trader does
9
The assumption of infinite mass of traders and limx−→Q α(x) = ∞ together guarantee that it is always possible that the
price could exceed θ. Otherwise, when θ is high enough, a trader (with a high enough signal) will foresee that, after traders
have bought all of the asset from the passive agent, or after all traders have bought the asset to their maximal capacities,
the price will be still below θ, then this trader will never sell the asset, while a trader with a low signal is not sure about this
so she sells. The equilibrium strategy in this case will be much more complicated and the result that the asset price is stuck
somewhere below its publicly recognized fundamental value simply because we are running out of asset supply or running
out of buyers is unrealistic.
10
When θ < η2 , some traders will receive negative signals, which is perfectly compatible with the assumption that θ is
non-negative. When θ < η, those with v < η2 will have a truncated belief support because θ cannot be below zero. This
cause traders with very low signals different strategies, which will be characterized in Appendix D and G. The rest of this
section will ignore this special case.
7
not know how many others’ signals are lower or higher than hers. This is an important element in the
model because the lack of common knowledge about the belief of the population prevents traders from
perfectly coordinating each other. In contrast, in the standard literature with a common posterior belief,
perfect coordination rules out the existence of the bubble by backward induction. In addition, the above
specifications are such that all traders’ posterior belief have exactly the same shape except for a horizontal
shift. Hence, traders’ buying and selling strategies are dispersed because they have different beliefs, and
they behave the same relative to their respective signals which gives simple form of solutions.
To make traders enter the market gradually and steadily so that price path is smooth and strictly
increasing over time, we add the following market friction. Traders finance their purchases entirely by
borrowing, but loans are granted in a smooth stream over time (interest on loans is zero). If a trader
wishes to buy, she must wait until she is granted a loan. At each instant, traders who wish to buy are
uniformly randomly chosen and granted the loan, irrespective of their types (signals). Hence, traders with
loans enter the market sequentially and the price increases steadily. Note that this friction is not essential
in our model, and the bubble still exists without it
11
. Once chosen, a trader gets just enough of a loan
to buy her desired amount of shares. If she wants to sell, she can do so right away. All her transactions
are executed instantly at the current market price. The stream of loans is available long enough (everyone
who keeps requesting a loan will eventually get one) so that we do not need to worry that the price may
stop rising because loans are not available. Denote u(t) the rate of influx of loan (e.g. $ per second) at
the instant t, with u(t) > 0, ∀t ≥ 0, which is exogenous and publicly known.
After t = 0, because θ is above the initial price, traders’ purchases continuously drive up the price.
When the price is high enough and no one wants to buy any more, the price stops rising. Denote this
random stopping time T and the price pT . Given the strategy profile of all types of traders, pT is a function
of θ and everyone can perfectly deduce θ from pT when the price stops rising. If pT > θ at this moment,
then the existence of the bubble becomes a common knowledge. It is natural to assume that this triggers
the bursting of the bubble endogenously (if a bubble exists). This is in contrast to the endogenous bursting
in AB2003 where the crash is triggered by a threshold on accumulated sales. After that, the passive agent
is willing to buy only at price p = θ, i.e., the price is fixed at θ after T . All traders who still hold the asset
have to liquidate at price p = θ to the passive agent. In the end, no trader holds the asset and all shares
go back to the passive agent.
11
If traders enter the market all at once at the beginning, they push the price up to the peak instantly, and then the price
plummets back instantly. Though this infinitely fast price jumps look unrealistic, the bubble still exists.
8
Individual trader’s purchases and sales cannot be observed by others and only the price is publicly
observable. There is no noise in the price, so if sales can affect the price, then θ can be deduced from
the first strategic selling, given the strategy profile. To prevent the strategic selling from being inferred
from the price and uncertainty resolved prematurely, we assume that a trader’s sale is instantly absorbed
by other traders’ purchases, as long as there are traders willing to buy. Specifically, if a trader sells the
asset to the passive agent and repay the loan, this repayment instantly becomes a loan available to others
buyers. This internal loan transfer is not subject to loan control and thus is not part of the stream u(t).
These transactions take place instantly. Hence, u(t) is the net injection of loans into the trader population
at instant t, and the strategic selling is digested internally and is not reflected in the price12 .
Since all transactions are executed at the current price prescribed by equation (1), the price path is
determined by u(t). u(t) is known, hence the entire price path is fully anticipated13 , until the sudden crash.
In this model we only require the price to be strictly and continuously increasing over time, so there is
no restriction on u(t) as long as u(t) > 0, ∀t, and our model is flexible enough to replicate realistic price
dynamics. This is in contrast to AB2003 which relies on an exogenous exponential price growth, and to
Doblas-Madrid (2012) which requires exponentially growing endowments to support the price path.
A trader incurs a fixed transaction cost c each time she changes her position, irrespective of price or
shares traded14 . There is no proportional transaction tax, but there is a capital gains tax of rate τ . The
capital gains tax is levied when a trader has a positive profit that has realized (after deduct the fixed
transaction cost). A trader’s profit before tax is thus determined by her purchase prices and sale prices,
minus the transaction costs. With the capital loss tax credit, when a trader suffers a capital loss L after
sale, she is entitled a tax credit L. If she has a capital gain later and needs to pay tax, she will be exempted
a tax payment up to the amount of τ L. The capital loss tax credit is said to be perfect if there is no ceiling
on the credit and the credit can be carried backwards and forwards indefinitely to offset taxable profits,
and therefore the tax benefit τ L can be regarded as a profit. We will discuss the case with perfect tax
credit and the case without any tax credit.
12
Assume that the trading volume is not observable.
This can be seen from the following steps.R At any time t, the instantaneous increment of traders’ aggregate position
t
u(ν)
is u(t)
p(t) , and the aggregate position is Dr (t) = ν=0 p(ν) dν. Combine this equation with p(t) ≡ α(Dr (t)) and the boundary
condition p(0) = 0, we can solve
qforR p(t). See Appendix A for details. For example, when p(t) = ρDr (t) (i.e. α(·) is linear),
2 t
where ρ is a constant, p(t) =
ρ ν=0 u(ν)dν. Therefore, the price will keep increasing in a predetermined manner, and
there will be no deviation from the anticipated path until the bubble bursts.
14
The reason why we use a fixed transaction cost instead of a proportional transaction tax is that, with a proportional
tax, we no longer have simple form of solutions and we have not been able to solve for them due to the existence of inverse
of unknown functional Pb∗−1 (·).
13
9
To rule out the possibility that some types of traders never stop buying so that the bubble grows
forever, we assume that the size of the bubble has an upper bound B. Once p(t) − θ > B, the bubble
bursts exogenously. We are only interested in the case of endogenous burst. We will show that in the
equilibrium with endogenous bursting the size of the bubble size is finite and constant so that, if B is large
enough, B is never binding. In addition, we assume that all traders selling exactly at the instant when the
bubble bursts receive the full pre-crash price.
Figure 2 shows a simple example of the dynamics. In each panel, traders’ signals are continuously
distributed between θ − η2 and θ + η2 along the horizontal axis, given θ, while the vertical axis measures the
mass of traders (at each signal v). The shaded area is the measure of traders who hold the asset. If we
assume that everyone who has bought has the same number of shares (which will turn out to be the case
in equilibrium), the shaded area is also proportional to Dr (t). In panel (a) at t = 0, everyone wants to buy
but none of them have bought any shares yet. In panel (b), some traders (of all types) receive loans and
become shareholders, and all others (all types) are seeking to buy. In panel (c), the price is high enough
such that low types no longer want to buy but simply hold (the shaded area of “hold” does not rise any
more), while the rest are still seeking to buy. In panel (d), the price is even higher such that not only
“no-buy” spreads to higher types of traders, but low types now start to sell. However, there are still types
who wish to buy. Recall that each type has an infinite number of traders waiting to enter the market. In
panel (f), the “no-buy” finally reaches the highest type, θ + η2 , who stops buying. As a result, no one buys
any more and the price stops rising and the bubble bursts. Note that in Figure 2, we have assumed that
all traders use a trigger strategy, where a trader will not restart buying once she has stopped buying and
she will never re-enter the market once she has sold. This strategy will turn out to be the equilibrium
strategy.
3
Preliminary analysis
In this section, we will define the equilibrium, impose two technical assumptions on traders’ strategies and
show that the dynamic game can be simplified to a static-like game by reducing traders’ strategy space.
We will establish that in equilibrium traders use trigger strategies (Proposition 3.1) and their decisions are
strictly increasing (Lemma 3.1). Readers who are not interested in these details can jump to Section 4.
Definition 3.1. A trading equilibrium is a Perfect Bayesian Nash equilibrium in which traders hold the
(correct) belief: whenever a trader v is not seeking to buy (temporarily or permanently), she (correctly)
10
number of
traders
buy
buy
buy
hold
θ−
η
2
θ+
trader
type
η
2
θ−
η
2
θ+
(a) t = 0
θ−
η
2
θ+
(b)
buy
η
2
(c)
buy
hold
hold
hold
sell
θ−
η
2
sell
η
2
θ+
η
2
θ−
(d)
sell
η
2
θ+
(e)
η
2
θ − η2
θ + η2
(f) t = T , bubble bursts
Figure 2: Dynamics of the mass of shareholders
believes that all traders with signal equal to or smaller than v are not seeking to buy.
This definition imposes a natural assumption on traders’ equilibrium beliefs, without which it will be
difficult to characterize an equilibrium.
With the positive fixed transaction cost c, a trader will trade only finite number of times. Due to the
linearity of the problem and the cost c, it is optimal for traders to either hold the maximum long position
or does not hold any asset at all. At any given price, a trader’s asset value is linear in her position.
With the transaction cost, if buying is profitable, then it is optimal to buy to the maximum long position;
conversely, if selling is profitable, then it is optimal to sell all shares15 . Hence, a traders’ position space in
equilibrium is reduced to {0, 1}.
Because we assumed simple linear tax rate, the profit or loss realized after sale (whether in current tax
year or previous years) and realized tax payment and benefit are all sunk and will not affect decisions. The
trading history affects decisions only when a trader is currently holding the asset and whether she can sell
before the crash is uncertain, i.e. her most recent purchase price may affect her selling decision. At any
given price, a trader has three options: seeks to buy to the maximum long position (buy), does not change
15
See Appendix A and proof of Lemma 1 in AB2003 for details.
11
her current position (hold ), and sells all her shares (sell ). Let A(p, v, h, Pp ) denote the strategy at price p of a
trader v with current position h ∈ {0, 1} and the most recent purchase price Pp . Pp is a random variable and
is relevant only when h = 1. Then A is defined on [0, ∞) × [− η2 , ∞) × {0, 1} × [0, ∞) → {buy, hold, sell}.
Definition 3.1 immediately implies the following corollary, which states that when trader v’s action is
not buy, the actions of all types weakly lower than trader v must not be buy, and when she is buying, all
types weakly higher than her are still buying, irrespective of trading histories or current positions. This is
because the belief in Definition 3.1 must be correct for all types of traders.
Corollary 3.1. A(p, v, h, Pp ) 6= buy =⇒ A(p, v 0 , h0 , Pp0 ) 6= buy, ∀v 0 ≤ v and ∀p, h, Pp , h0 , Pp0 ;
A(p, v, h, Pp ) = buy =⇒ A(p, v 0 , h0 , Pp0 ) = buy, ∀v 0 ≥ v and ∀p, h, Pp , h0 , Pp0 .
Corollary 3.1 implies that traders’ strategies are symmetric in the sense that traders with the same
signal are either all buying or all not buying.
Now we can formally define the bursting price pT :
η
η
pT = inf {p|A(v, p, h, Pp ) 6= buy, ∀v ∈ [θ − , θ + ], ∀h and ∀Pp }
2
2
That is, the bubble bursts when no one wants to buy.
Let Pb∗ (v) denote the price at which the action of trader v is not buy for the first time. Let Ps∗ (v, Pp )
denote the price at which trader v sells her shares for the first time, given her purchase price Pp (since her
position is either 0 or 1 and this is the first-time sale, Pp is well defined.). By definition, Pp < Pb∗ (v) ≤
Ps∗ (v, Pp ), ∀v and ∀Pp 16
To derive an equilibrium, we will need two technical assumptions:
Assumption 3.1. Pb∗ (v) is continuous in v, ∀v and differentiable ∀v > η2 , and Ps∗ (v, Pp ) is continuous in
both v and Pp .
Lemma 3.1. Pb∗ (v) is strictly increasing in v, ∀v ≥ η2 .
Then the inverse function Pb∗−1 (v), ∀v ≥ η2 , is well defined.
Assumption 3.2. A trader cannot switch actions more than once in one instant.
Then we have the following proposition.
Pb∗ (v) < Ps∗ (v, Pp ) is because receiving the loan and becoming able to buy the asset does not involve any new information,
a trader’s belief conditional on having received the loan and bought the asset must be the same as that conditional on having
not received the loan. Hence it should not happen to any trader that, if she has never bought the asset she prefers buying
at a price, while if she has bought the asset she prefers to sell at a lower price.
16
12
Proposition 3.1. (Trigger-strategy): In equilibrium, if a trader holding no shares stops buying, she will
never restart buying again. If a trader has sold her shares, she will never return to the market.
Proposition 3.1 implies that, when the highest type θ +
η
2
decides to stop buying for the first time at
Pb∗ (θ + η2 ), all other types have already done so. Since no one is buying at this moment, the bubble bursts
at pT = Pb∗ (θ + η2 ). Then θ = Pb∗−1 (pT ) − η2 , and this is how everyone can perfectly infer θ from the bursting
price pT .
Proposition 3.1 further reduces a trader’s strategy space to {Pb∗ (v), Ps∗ (v, Pp )}. Right after t = 0, if the
price ever rises above zero, then at least some traders want to buy, and some of them have received the
loan and bought to the maximum position 1, and others are waiting for the loan. If a trader v has bought
the asset at price Pp , she holds and waits until the price rises to Ps∗ (v, Pp ) then she sells all her shares. If
she has not received the loan, then at price Pb∗ (v) she stops trying and gives up. In either case, she will
never restarting buying again. Readers can review the price dynamics under this strategy profile in Figure
2.
4
Reduced form game
After T , the price jumps to θ and stays there thereafter. If a trader v sells before the crash, she gets the
selling price. Otherwise, she gets the post-crash price θ. Now we solve the individual trader’s problem
backward, starting with the sale decision. In what follows we will focus on traders with v >
strategies of those with v ≤
4.1
η
2
η
.
2
The
will be characterized in Appendix D and G.
Sale decision
Consider a trader v who has bought at price Pp and plans to sell at Ps . A trader is taxed only if she has a
positive profit, i.e. Pp + 2c < Ps (sell before the crash) or Pp + 2c < θ (sell after the crash) and not taxed
otherwise. If there is a perfect tax credit, a trader also receives a benefit τ (Pp + 2c − θ) due to the tax
credit when she incurs a loss Pp + 2c − θ, where Pp + 2c > θ17 .
When a trader’s purchase price Pp is low enough such that Pp + 2c < Pb∗−1 (Ps ) − η2 , it is possible that
the post-crash sale is still profitable and taxable (when θ > Pp + 2c). In this case, given that all other
17
In equilibrium, pre-crash sales always give positive profits. A pre-crash sale is a scheduled sale. If a trader buys at Pp
and sells at planned Ps and incurs a loss, then he will not buy at Pp .
13
traders stop buying at Pb∗ (·), the expected payoff is
Z
"Z
Pp +2c
[θ−Pp −2c+τc (Pp +2c−θ)] φ(θ|v)dθ+(1−τ )
θ=Pb∗−1 (Pp )−η2
Pb∗−1 (Ps )−η2
(θ−Pp −2c)φ(θ|v)dθ
θ=Pp +2c
η i
+(Ps −Pp −2c) 1−Φ(Pb∗−1 (Ps )− |v)
2
h
#
(2)
where τc equals τ when there is a perfect tax credit and 0 when there is no tax credit. In the first integral
the post-crash sale incurs a loss and is not taxable, and there is a tax benefit; in the second integral the
post-crash sale is profitable and taxable. The third term is the expected payoff from selling before the
crash which is also taxable.
When Pp is high enough such that Pp + 2c > Pb∗−1 (Ps ) − η2 , in which case the post-crash sale always
incurs a loss and is non-taxable, the expected profit is
Pb∗−1 (Ps )− η2
h
η i
[θ−Pp −2c+τc (Pp +2c−θ)] φ(θ|v)dθ+(1−τ )(Ps −Pp − 2c) 1−Φ(Pb∗−1 (Ps )− |v)
2
θ=Pb∗−1 (Pp )− η2
Z
(3)
Let us call the case with perfect tax credit case C, and let ωC (Pp , Ps ) denote the expected payoff from
having already bought at Pp and planing to sell at Ps in this case. In case C, where τc = τ , equation (2)
and (3) can be rearranged and both be written as
"Z ∗−1
Pb
Case C :
ωC (Pp , Ps ) = (1 − τ )
(Ps )− η2
h
η i
θφ(θ|v)dθ + Ps 1 − Φ(Pb∗−1 (Ps ) − |v) − Pp − 2c
2
θ=Pb∗−1 (Pp )− η2
#
(4)
When there is no tax credit, let us call the case in (2) case T , and the case in (3) case N T , and let
ωT (Pb , Ps ) and ωN T (Pb , Ps ) denote the expected payoff from having already bought at Pp and planing to
sell at Ps in case T and N T , respectively. Then equation (2) can be written as
"Z ∗−1
Z
Pp +2c
Case T :
ωT (Pp , Ps )
=
[θ−Pp −
θ=Pb∗−1 (Pp )−η2
Pb
2c] φ(θ|v)dθ+(1−τ )
(Ps )−η2
(θ − Pp − 2c)φ(θ|v)dθ
θ=Pp +2c
η i
+ (Ps −Pp − 2c) 1−Φ(Pb∗−1 (Ps )− |v)
2
h
#
(5)
and (3) can be written as
Z
Case N T : ωN T (Pp , Ps ) =
Pb∗−1 (Ps )− η2
[θ−Pp −
θ=Pb∗−1 (Pp )− η2
2c] φ(θ|v)dθ+(1−τ )(Ps −Pp − 2c)
h
η
1−Φ(Pb∗−1 (Ps )− |v)
i
2
(6)
Without the tax credit, a trader must be either a case T trader or a case N T trader. Since Pp ∈ [0, Pb∗ (v)],
if Pb∗ (v) + 2c < Pb∗−1 (Ps∗ (v)) − η2 in equilibrium, then it is impossible that Pp + 2c > Pb∗−1 (Ps∗ (v)) − η2 . Then
N T trader does not exist and all traders are case T traders and use (5) to evaluate their expected payoff. If
14
Pb∗ (v)+2c > Pb∗−1 (Ps∗ (v))− η2 in equilibrium, then those buying at prices such that Pp +2c > Pb∗−1 (Ps∗ (v))− η2
are case N T traders and use (6), and the rest are case T traders and use (5). It will turn out that both
cases, Pb∗ (v) + 2c ≷ Pb∗−1 (Ps∗ (v)) − η2 , can be supported in equilibrium (by different parameter values).
Let ωX (Pp , Ps ) denote the expected payoff in the three cases, where X ∈ {C, T, N T }. A trader v faces
a sale decision
max ωX (Pp , Ps )
(7)
Ps
For a trader who considers selling at a current price Ps , if
∂ωX
∂Ps
∂ωX
∂Ps
> 0, then she should not sell; when
≤ 0, then she should sell. Differentiating ωX (Pp , Ps ) w.r.t Ps and setting
∂ωX
∂Ps
= 0 gives the first order
condition (write all three cases in one equation):
h
i
φ(Pb∗−1 (Ps ) − η2 |v)
η
1
∗−1
1 − τ = (1−τ )(Ps −Pp −2c)−(1−τct )(Pb (Ps )− −Pp −2c)
0
2
Pb∗ (Pb∗−1 (Ps )) 1−Φ(Pb∗−1 (Ps )− η2 |v)
(8)
where τct equals τ in case C and T , and equals 0 in case N T . This first order condition can be interpreted
in terms of marginal benefit and cost. For a trader who evaluates selling at Ps vs. Ps + ∆ (∆ is small),
the relevant marginal event is that the bubble bursts between Ps and Ps + ∆. On the left hand side, the
marginal benefit of selling at Ps + ∆ over Ps is (1 − τ )∆ (the after-tax price appreciation). On the right
hand side, the marginal cost is that she could get caught in the crash if the bubble bursts in between Ps
and Ps + ∆, which equals the profit difference multiplied by the probability of bursting between Ps and
Ps + ∆. In the brackets is the profit difference between pre and post-crash transactions, i.e. the loss due
to crash. The first term in the brackets is the pre-crash profit which is always positive and taxable, and
the second term is post-crash payoff which is positive and taxable in case T and is negative in case N T .
Outside the brackets is the hazard rate of bursting at Ps . Dividing both sides by ∆ and letting ∆ → 0,
we have equation (8).
4.2
Purchase decision
Given that all other traders stop buying at Pb∗ (·) and trader v will sell at Ps∗ (·, ·), we solve for trader v’s
optimal choice of stop-buy price, Pb . By Corollary 3.1 the bubble will not burst below Pb (v) for trader v.
When trader v evaluates the expected profit from buying at the current price Pp and selling at at Ps∗ (v, Pp ),
the expected profit is ωX (Pp , Ps∗ (v, Pp )). It is not difficult to verify that the higher the purchase price, the
lower the expected profit, i.e.
∂ωX
∂Pp
< 0. Then as long as ωX (Pp , Ps∗ (v, Pp )) > (1 − τ )R, buying the asset
at Pp is preferred to buying the outside option, and a trader will stop trying to buy the asset only when
15
ωX (Pp , Ps∗ (v, Pp )) ≤ 0. Hence the the optimal stop-buy price Pb for trader v is defined by
ωX (Pb , Ps∗ (v, Pb )) = (1 − τ )R
(9)
In equilibrium Pb∗ (v) = Pb (v). Recall that the crash is triggered by stop-buy decisions (of the highest type
of traders). If a trader (who has just received a loan) decides to buy at the current price, then she is no
longer a concern to others because all that matters is the stop-buy decisions of those who have not bought
yet. Call a trader the break-even trader if she has not bought yet and finds that buying at current price
Pp and selling at Ps∗ (·, Pp ) gives an expected profit equal to (1 − τ )R (equation (9)) so that she decides to
stop buying at Pp . Only the break-even traders are important to solving the problem. So the equilibrium
is defined by equations (7) and (9).
5
Equilibrium with perfect tax credit
Proposition 5.1 shows that, when there is a perfect tax credit and if
2c+R
η
< 12 , i.e., when c and R are small
relative to η, then there is a unique trading equilibrium with a positive bubble, which corresponds to an
interior solution of equation (7). Proposition 5.2 shows that if
2c+R
η
≥ 12 , the trading equilibrium has no
bubble18 , which corresponds to a corner solution of (7).
Proposition 5.1. (Tax credit with bubble) Suppose there is a perfect capital loss tax credit. When
p
2c+R
< 12 , there exists a unique trading equilibrium in which the bubble size is B = η − (4c + 2R)η > 0,
η

η p
∗

buy,
if
price
<
P
(v)
=
v
+
− (4c + 2R)η;

b

2

η
η
and the bubble bursts at θ + B, and a trader v > will
sell, if price ≥ Ps∗ (v) = v + ;

2
2



∗
∗
hold, if Pb (v) ≤ price < Ps (v).
For traders with v ≤ η2 , their strategies are more complicated but less important and are relegated to
Appendix D. The equilibrium strategies (including strategies of traders with v ≤ η2 ) are depicted in Figure
3. In panel (a) of Figure 3, when c and R are very small, even the lowest type v = − η2 buy and sell. In
√
panel (b) when 2 − 3 ≤ 2c+R
< 12 , some of the traders with v ≤ v C < η2 never buy the asset, since their
η
stop-buy prices Pb∗ (v) ≤ 0. v C is defined in Appendix D.
Proposition 5.1 prescribes that with a perfect tax credit and when 2c+R
< 21 , a trader v > η2 always
η
p
stops buying at Pb∗ (v) = v + η2 − (4c + 2R)η if she has not bought yet, and sells at Ps∗ (v, Pp ) = v + η2 if
she already bought. The most important result is that the size of the bubble B is not affected by capital
18
It has a negative “bubble”.
16
Ps∗ (v)
p
p
Ps∗ (v)
v+
v+
η
2
Pb∗ (v)
v+
η−
η−
p
η
2
η
2
−
p
(4c + 2R)η
Pb∗ (v)
p
(4c + 2R)η
v+
η
2
−
p
(4c + 2R)η
(4c+2R)η−2c−R
− η2
η
−
2
η
2
p
(4c+2R)η
(a)
2c+R
η
<2−
√
v
3
(b) 2 −
√
3<
v
η
2
vC
2c+R
η
<
1
2
Figure 3: Equilibrium strategies under perfect tax credit and positive bubble
gains tax. In a nutshell, the tax credit provides a compensation to the loss in crash and neutralizes the
repressive effects of the tax on traders’ strategies. We will explain the intuition in more detail in Section
6 by comparing the case C, T and N T with the tax-free case.
In contrast, the fixed transaction cost and the outside option can reduce the bubble ( ∂B
< 0 and
∂c
∂B
∂R
< 0). When c or R increases, the minimal margin between buying and selling prices for a trader to
make a profit is larger, which lowers the stop-buy price and therefore deflates the bubble. Conversely, when
c or R decreases, buying the asset at higher prices and incurring the cost are still preferable to receiving
R (i.e. the minimum margin between purchase and selling prices narrows), and hence generates a larger
bubble.
Our model also shows that when c and R are small, they have very large marginal effect on the bubble.
√
= −∞. The reason is as follows.
Let b ≡ 2c+R, then the size of the bubble is B = η − 2bη, and limb→0 ∂B
∂b
Because sale prices are not affected by c or R, bubble size is determined by Pb∗ (·). A trader’s stop-buy
√
√
price is always 2bη lower than her selling price, so 2bη can be regarded as the effective transaction
√
cost. The larger the gap 2bη between the stop-buy and sale prices, the smaller the bubble. When there
is no risk of crash at all, b is a minimal margin between purchase and selling prices for a trader to buy
the asset instead of the outside option. Since it is possible that the bubble bursts in between this gap, a
trader needs to lower the purchase price and widen this profit margin to compensate for the risk. Larger
gap means higher probability of bursting in between, which requires an even larger margin. But Pb∗ (·)
17
will not be shifted downwards infinitely, since bubble size is also decreasing as Pb∗ (·) decreases, and the
loss of being caught in the crash diminishes. There is a stop-buy price where the expected profit equals
the outside option. As a result, a small fixed transaction cost or outside option can induce a large gap
between the stop-buy and sale prices, and can effectively reduce the bubble, although the marginal effect
is decreasing. As can be seen from the explanation, this result does not rely on our choice of specific belief
distributions. Figure 4 shows the relationship between the normalized size of the bubble
B
η
and ηb .
B
η
1
1
b
η
Figure 4: Relationship between
B
η
and
b
η
The equilibrium bubble size B never exceeds η. B is also increasing in η, which means that the more
dispersed the public opinion about the asset fundamental value, the larger the bubble size. B actually
can be made arbitrarily large if we let η −→ ∞, and c and R be arbitrarily small. Conversely, if we can
effectively reduce the belief dispersion η, the size of the bubble will decrease.
When θ ≥ η2 , upon the bursting, traders with signals in the lower range [θ − η2 , θ +
η
2
−
p
(4c + 2R)η]
have fled the market successfully with positive profit, while the rest (v in the higher range [θ + η2 −
p
(4c + 2R)η, θ + η2 ]) are caught in the crash. Even with this lowest possible realization of θ, which
generates signals distributed on [− η2 , η2 ], there are traders buying. So there is always a bubble and a crash
when
2c+R
η
< 12 , no matter what is the realization of θ.
For a trader v, the highest possible realization of θ is v + η2 . If the price rises above v + η2 , then she
knows for sure that a bubble exists. Upon the bursting, the price is pT (θ) = θ + B and traders with
p
v ∈ [θ − η2 , θ + η2 − (4c + 2R)η] are sure that a bubble already exists, and they are exactly those who
have fled the market before the crash.
If τ = 0 and c = R = 0, then Pb∗ (v) = Ps∗ (v, Pp ) = v + η2 and B = η. In this case the stop-buy strategy
equals the selling strategy.
The following proposition states that when the transaction cost c and the return from outside option
R are large, or traders’ opinions are sufficiently concentrated (η is small), there is a different equilibrium
where there is no bubble (bubble size < 0).
18
Proposition 5.2. (Tax credit without bubble) Suppose there is a perfect capital loss tax credit. When
2c+R
η
≥ 12 , there exists a “unique” trading equilibrium in which the bubble size B =
η
2
− 2c − R ≤ 0, and a
trader
• v ≤ 2c + R will never buy the asset;



buy, if price < v − 2c − R;


• v > 2c − R will
hold, if v − 2c ≤ price < v + η2 .



 sell, if price ≥ v + η
2
In this equilibrium, when the highest type stops buying, the price is still lower than θ. When the
“bubble” bursts, the price jumps up to θ. This is because the fixed transaction cost and the return from
outside option are too large compared to η, and the stop-buy strategy is pushed so low that the bubble
becomes negative. With a negative “bubble”, if a trader sells before the “crash”, she forgoes the price
appreciation that would have certainly realized had she waited till the “bubble” bursts. As a result,
everyone has an incentive to hold the asset until the uncertainty is resolved. Obviously the strategy profile
in Proposition 5.1 is no longer an equilibrium. Knowing that she will be selling at θ after the “crash”
and the expected sale price is E[θ|v], a trader v should buy the asset if the current price is lower than
E[θ|v] − 2c − R. Notice that when θ ≤ 2c + R − η2 , even the highest type θ + η2 stops buying at price zero,
thus no one buys from the beginning and the price stays at zero throughout. This is the case where the
asset fundamental has a small rise, but the price does not reflect it at all because transaction cost or the
return from outside option is too large compared to the dispersion of traders’ opinion.
This equilibrium is unique in the sense that the stop-buy strategy and “bubble” size are unique, but
selling strategy is not unique. For trader v, since the bubble will burst below v +
η
2
for sure and the price
will be fixed at θ thereafter, price will never rise above v + η2 . So all strategies selling at any price above
v+
6
η
2
are optimal, although they will never be implemented.
Equilibrium without tax credit
When break-even traders are N T (T ) traders, we call the bubble an N T (T ) bubble.
Proposition 6.1. (No credit but with bubble) Suppose that there is no capital loss tax credit.
1. N T bubble: When τ <
1
2
−
2R+4c
η
(1− 2c
)2 − 4R
η
η
and
R
η
< 18 (1 −
trading equilibrium in which the bubble size is B =
19
η
2
2c 2
)
η
−
c
η
and
c
η
<
√
3−2 2
,
2
there exists a unique
+ DN T > 0, and the bubble bursts at θ + B, and
a trader v > η2 will


buy, if price < Pb∗ (v) = v + DN T ;




h
i


η
1−τ
τ
1
2c


v−
P
+
(1−τ
)(
−τ
−τ
)+τ
d


NT ,


1−2τ p
1−2τ
2
η
 1−2τ

sell, if price ≥ Ps∗ (v, Pp ) =
if Pp > v−2c−DN T ;







 v + η , if P < v − 2c − D

p
NT

2



 hold, if P ∗ (v) ≤ price < P ∗ (v, P ).
p
s
b
h
i
q
2R+4c
where DN T ≡ η 21 − τ − τ 2c
−
d
and
d
≡
(1 − 2τ )(1 − τ ). In particular Ps∗ (v, Pb∗ (v)) =
NT
NT
η
η
h
i
2τ
v + η 12 − τ − τ 2c
+
d
. All the break-even traders are N T traders.
η
1−2τ N T
2. T bubble: When
τ<
1− 4c
− 2R
η
η
(1− 2c
)2 −2 R
η
η
1
2
−
2R+4c
η
(1− 2c
)2 − 4R
η
η
<τ <
and 81 (1− 2c
)2 − ηc <
η
R
η
1− 4c
− 2R
η
η
(1− 2c
)2 −2 R
η
η
and
< 12 (1− 4c
) and
η
c
η
R
η
< 18 (1 −
2c 2
)
η
−
c
η
and
c
η
<
√
3−2 2
,
2
or when
< 14 , there exists a unique trading equilibrium
η
η
in
 which bubble size is B = 2 + DT > 0, and the bubble bursts at θ + B, and a trader v > 2 will
 1+4τ c −2d


buy, if price < Pb∗ (v) = v + DT ;
T

η


η, if τ 6= 12
2(1−2τ )
η
∗
sell, if price ≥ Ps (v, Pp ) = v + 2 ; , where DT ≡
 −c − 2c+R

, if τ = 12 .


1+ 2c
 hold, if P ∗ (v) ≤ price < P ∗ (v, P ).
η
p
s
rb
h
i
2 + 2R (1 − 2τ ) + 4c . All the break-even trader are T traders.
)
and dT ≡ (1 − τ ) τ (1 − 2c
η
η
η
For traders with v ≤ η2 , their strategies will be characterized in Appendix G.
The equilibrium strategies with an N T bubble in Proposition 6.1 is depicted in Figure 5 (which also
depicts strategies for v ≤ η2 ). This part is derived when the break-even traders are in case N T , i.e. their
post-crash sales (if they purchase) are non-profitable for sure and they use ωN T to evaluate their expected
profit. When τ , c and R are very low, only a small profit margin is required and the stop-buy price is
close to the selling price. With such small margin, however, a break-even trader cannot make any profit if
caught in the crash, even if θ turns out to be high (θ = v + η2 ), which makes her an N T trader. Actually,
not only break-even traders are N T traders, but all traders buying above the dividing line v − 2c − DN T
are N T traders, while those buying below the dividing line are T traders. For N T traders, since the tax
is effectively imposed only on pre-crash transactions and there is no tax credit, in the first order condition
the Ps∗ (v, Pp ) does depends on purchase price Pp . Hence different purchase prices lead to different selling
prices, and these selling prices have an upper bound v + η2 and a lower bound Ps∗ (v, Pb∗ (v)). For T traders
(buying below the dividing line), it is still possible that her post-crash sale is profitable so that she uses ωT
to evaluate her expected profit. In particular, a T trader always sells exactly at v +
irrespective of her purchase price.
20
η
2
(the upper bound),
p
Ps∗ (v, Pb∗ (v))
Pb∗ (v)
v+
η
2
η
2
v + DN T
+ DN T
v − 2c − DN T
η
+DN T −2c−R
2
− η2
Ps L
P sH
vN
vN
T
T
N oT
vN T
v
η
2
Figure 5: Equilibrium strategies without tax credit and with positive bubble in case N T
The bubble size is B =
η
2
+ DN T , and it can be verified that the B is decreasing in tax rate, fixed
transaction cost and outside option payoff, i.e.
∂B
∂τ
< 0,
∂B
∂c
< 0 and
∂B
∂R
< 0.
As break-even traders are N T traders, the tax has a distortion effect which moves selling strategies down below the (undistorted) strategies in case C. In the first order condition (equation (8)), the
relevant event is that the bubble bursts just before selling at Ps , i.e. θ = Pb∗−1 (Ps ) − η2 . The first
φ
=⇒ 1 =
order condition in case N T is therefore 1 − τ = [(1 − τ )(Ps − Pp − 2c) − (θ − Pp − 2c)] P1∗0 1−Φ
b
h
i
φ
p −2c
1
(Ps − Pp − 2c) − θ−P
, while if there is no tax the first order condition is 1 = [(Ps − Pp − 2c) − (θ − Pp
0
1−τ
P ∗ 1−Φ
b
Hence the payoff difference between pre and post-crash in case N T is larger than the difference without
tax19 , which advises that traders sell more cautiously and conservatively, and all N T traders sell below
the undistorted strategy v + η2 .
h
In comparison, when there is a perfect tax credit, the first order condition is 1−τ = (1−τ )(Ps −Pp −2c)
i
φ
−(1 − τ )(θ − Pp − 2c) P1∗0 1−Φ
, which, after normalization, is exactly the same as without tax. It is clear
b
that the tax credit provides a compensation, which scales the loss θ − Pp − 2c by a factor 1 − τ . Thus
the tax credit reduces the payoff difference between pre and post-crash and restores it to its tax-free level.
This encourages traders to sell aggressively as if there is no tax.
Back to case N T , a trader’s stop-buy price is also lower than the tax-free stop-buy price. Combine
(6) and (9), we find that positive profits are all scaled by a factor 1 − τ , but the negative ones are not,
which implies that a trader has to lower her purchase price (stop buying earlier) to maintain the minimum
19
Recall that θ − Pp − 2c < 0.
21
reservation profit (1 − τ )R. This is a direct effects of capital gains tax on the stop-buy strategies. The
direct and indirect effect of capital gains tax together depress the bubble so that it is smaller than the
tax-free bubble. In comparison, when there is a perfect tax credit, both direct and indirect effects are
eliminated and the tax fails to deflate the bubble.
Also we observe that the selling price Ps∗ is reversely related to purchase price Pp for N T traders.
This is because the marginal purchase cost of the pre-crash sale in the first order condition is scaled by
factor (1 − τ ): when purchase cost Pp + 2c increases by ∆, tax payment reduces by τ ∆, so the marginal
purchase cost in the pre-crash sale only increases by (1 − τ )∆, whereas the purchase cost of the post-crash
sale does not enjoy this benefit. This implies that the tax, which is effectively imposed only on pre-crash
sales, lowers the purchase cost in the pre-crash case compared to the post-crash case and makes the former
relatively more attractive, and the larger the purchase cost (purchase price), the more attractive the former
is. Hence, given a signal v, the higher the purchases price, the stronger an N T trader prefers to selling
before the crash by selling earlier at a lower price. In particular the break-even traders’ who have the
highest purchase prices (if they have purchased) sell the lowest (earliest) at Ps∗ (v, Pb∗ (v)).
The equilibrium strategies with a T bubble in Proposition 6.1 is depicted in Figure 6. The difference is
p
p
Ps∗ (v, Pb∗ (v))
v+
η
2
Ps∗ (v, Pb∗ (v))
v − 2c − DT
v − 2c − DT
Pb∗ (s)
v+
η
2
Pb∗ (v)
v + DT
v + DT
η
2
+ DT
η
+DT −2c−R
2
− η2
v
η
P L
vTDiv 2
vTPs H vT s
vTN oT
(a) Lowest traders sell together
vTPb
η
2
v
(b) Lowest traders never buy
Figure 6: Equilibrium strategies without tax credit and with positive bubble in case T
that, the break-even traders are now in case T . Because the tax and transaction cost are higher now, the
stop-buy price Pb∗ (v) = v + DT is now below the dividing line v − 2c − DT . Hence, not only break-even
traders, but all traders (v > η2 ) traders are T traders and there is no N T trader in this case. They buy at
22
such low prices that, if caught in the crash, it is possible that the post-crash sales are still profitable, even
for the break-even traders (if they buy). As such, they all use ωT to evaluate their expected profit and sell
at v + η2 , irrespective of their purchase prices.
The bubble size is B =
η
2
+ DT , and it can be verified that the it is decreasing in tax rate, fixed
transaction cost and outside option payoff, i.e.
∂B
∂τ
< 0,
∂B
∂c
< 0 and
∂B
∂R
< 0.
In case T the selling price is undistorted, compared with the tax-free case, because in the marginal
event the post-crash transaction is still profitable and taxable, just as the pre-crash transaction. Hence
the normalized first order condition is the same as the tax-free first order condition. But the stop-buy
price is again lower than its tax-free level, because the direct effect of the tax is still at work, which lowers
the stop-buy strategies directly. In comparison, when there is a perfect tax credit, the direct effect is
eliminated and the tax again fails to deflate the bubble.
Compare Figure 5 and 6, we see that when τ , c and R are small, break-even traders (N T traders) sell
lower than when τ , c and R are large (break-even traders are T traders). This is because in the former case,
their stop-buy prices are high, which cause their post-crash sales unprofitable for sure. So their selling
strategies are distorted downwards by the tax. In the latter case, low stop-buy prices make profitable
post-crash sales possible, which leaves selling strategies undistorted. Because the bubble size in our model
is determined by the stop-buy prices, the bubble is always larger in case N T than in case T .
Figure 7 depicts the partitioning of the parameter space. Panel (a) is in the
N T bubble is in the lower left corner where both
bubble where
-τ -
R
η
c
η
and τ are larger. When
c
η
c
η
c
η
- τ space with R = 0.
and τ are small, and T bubble area surrounds N T
and τ are even larger, there is no bubble. Panel (b) is in the
space. N T bubble is in the lower left corner where ηc , τ and
R
η
c
η
are all small, and T bubble area is
in between the two surfaces. Outside the larger surface there is no bubble. From the fact that bubble size
is decreasing in ηc , τ and
R
η
N T and T areas (when τ =
in both N T and T bubbles, and that DN T = DT on the boundary between
1
2
−
2R+4c
η
(1− 2c
)2 −4 R
η
η
), we know that without tax credit bubble size in case N T is
always larger than that in case T , and bubble size is decreasing in ηc , τ and
R
η
across the boundary. This
is depicted in Figure 8 (set R = 0). In the dark red part on the top are N T bubbles, and in the light blue
part in lower area are T bubbles.
Proposition 6.2. (No tax credit no bubble) Suppose there is no capital loss tax credit. When t ≥
1− 4c
− 2R
η
η
(1− 2c
)2 −2 R
η
η
, or
R
η
≥ 12 (1 −
4c
),
η
or
c
η
≥ 14 , there exists a “unique” trading equilibrium in which bubble size
N
B = η + DN
≤ 0, and a trader
23
τ
1
τ=
1− 4c
η
(1− 2c
)2
η
t
1
2
τ=
1
2
−
T bubble
No bubble
4c
η
(1− 2c
)2
η
R
__
N T bubble
3 √
− 2
2
1
4
1
2
h
c
η
c
__
h
(a) In the
c
η
(b) In the
- τ space (R = 0)
c
η
-τ -
R
η
space
Figure 7: Partitioning of the parameter space without tax credit
Figure 8: Bubble size (R = 0)
N
• v ≤ −DN
−
η
2
will never buy the asset;


 v + η + DN , if 2 R ≤ 1;


N

2
η
NN

buy,
if
price
<
P
(v)
≡

b

 v − 2c − R, if 2 R > 1

η
N
• v > max( η2 , −DN
− η2 ) will
η


hold, if PbN N (v) ≤ price < v + 2 .




 sell, if price ≥ v + η
2
24
N
where DN
≡ − τη − 2c +
η
τ
q
(1 − τ )(1 − 2τ Rη )
This equilibrium is unique, again, in the sense that the stop-buy price and bubble size are unique, but
selling strategy is not unique, by the same reason as in Proposition 5.2.
7
Empirical Predictions
7.1
Effects of Capital Gains Tax, Transaction Cost and Outside Option
The results of our model provides some insights for empirical research. First, the two main findings from
our model generate testable predictions. We showed that, the effect of capital gains tax on bubbles depends
on tax credit: when there is no tax credit (or the tax credit is imperfect), the size of the bubble is decreasing
in capital gains tax rate; when tax credit is close to perfect, the tax has little effect on bubbles. The bubble
size is decreasing in the transaction cost and the return from alternative investment opportunities, which
does not depend on tax policies. The equilibrium in our model is unique, so it should be able to test the
above results without worrying about the issue of multiple equilibria.
To demonstrate that our model can be empirically tested, we study several historical events of bubble.
We identify actual sizes of bubbles in these events and normalize them for comparison. This could potentially show how tax, cost and outside option affect bubbles. We also fit normalized actual bubble sizes to
theoretical results calculated from model parameters to test our model. But given the small sample size
here, the results showed is far from conclusive. Further empirical study is expected.
The four historical events of bubble we explore are the 1929 stock market crash, the Japan asset bubble
in the 1980s, the Internet Bubble in the late 1990s and the Bitcoin bubble in 2013. For each event of
bubble, we identify the following parameters: tax rate τ , transaction cost c, outside option R and belief
dispersion of investors (about the asset fundamental) η, from which we calculate the theoretical size of the
bubble. The tax credit of these events either falls in between perfect and no credit, or the information of
which is difficult to find, so we compare these events to theoretical results in both perfect and no credit
cases. In the case of no credit, we have to determine whether they are in the range of T or N T based on
the value of parameters.
In the real world there are so many factors that can affect asset bubbles (e.g. monetary policies,
credit control, exchange rates, tax policies, productivity, market capitalization) that comparing the sizes
of bubbles directly is essentially meaningless. To identify the effects of factors such as tax or interest rate
on bubbles, we have to normalize the actual sizes of bubbles before we can compare. Our model suggests
25
that the extent to which investors’ beliefs differ is an important factor that determines the size of the
bubble and is a candidate that can be used for normalization. However, the belief dispersion in a real
market is usually very difficult to identify. So we use the following method to infer this dispersion, η.
Rational traders’ belief in our model is distributed in the range [θ − η2 , θ + η2 ]. When the price rises to
Ps∗ (θ − η2 ) rational investors with lowest belief θ − η2 start to sell. In addition, the specification of our model
is such that the impact of this action to the price is perfectly concealed. The belief of investor in the real
market is not necessarily uniformly distributed or symmetrically distributed around θ, but arbitrageurs
(e.g. hedge funds) that hold significant shares and have various belief can be viewed as the rational traders
in our model. When the major arbitrageur with the lowest belief among them starts to sell, the price
impact should be felt by the market. Therefore we use the largest price drop in the data during the price
run-up20 as a proxy for the initial attack of rational traders with lowest belief, θ − η2 . In particular, we
treat the price right before the drop as Ps∗ (θ − η2 ) and calculate η by inverting Ps∗ (·).21 We also exclude
episodes of price drop that coincide with major exogenous shocks to the market.
We define the run-up of a bubble as a period of time up to the moment when the price hit the peak,
and the crash starts right after that. The actual size of a bubble is measured from the peak of the price
path to the average price within a certain period of time after the crash. This is because it usually took
time for actual crashes to reach the bottom and the price paths after the peak were also quite noisy. In
addition at the bottom there often existed downward price overshootings followed by slow recovery. The
average balances the fact that investors have opportunities to sell before the price hits the bottom and
the downward price overshooting which is usually lower than the true asset value. We set the length of
this period half of the length of the run-up. The actual path of indexes of these events are presented in
Figure 9, where in each panel the vertical dashed line is the cut-off between run-up and bursting and the
horizontal line is the average value of post-crash index.
The trading cost usually includes three components: broker’s commission, bid-ask spread and the market impact cost. For simplicity we only include broker’s commission, and we set c equal to the commission
rate times the asset price at the peak. For the outside option, R, we use the 3-month rate of return of the
treasury bond at 3 months before the crash multiplied by the asset price at the peak.
Using conditions set in Proposition 5.1 and 6.1, we can verify that there existed a bubble when assuming
1
We use the largest price drop during a period of time, which is scaled to 20
of the length of the run-up.
η
∗
The ex post identification of Ps (θ − 2 ) as such should not be viewed as traders being able identify true θ when the
bubble event has not fully unfolded, since one does not know whether this price drop is the largest or not before the final
crash, which has not realized yet.
20
21
26
5
4.5
5
4
3.5
4
3
2.5
3
2
2
1.5
1
1
0.5
0
1921−08−24 1923−07−27 1925−06−29 1927−05−24 1929−04−25 1931−03−26 1933−02−21 1935−02−01 1936−12−30
0
1982−01−01 1984−02−16 1986−04−02 1988−05−17 1990−07−02 1992−08−14 1994−09−29 1996−11−13 1998−12−29
(a) 1929 Stock Market Crash: S&P Stock Index
Source: CRSP
(b) Japan Asset Bubble: Nikkei 225 Stock Index
Source: Federal Reserve Bank of St. Louis
5.5
5
200
4.5
4
150
3.5
3
2.5
100
2
1.5
50
1
0.5
0
1997−01−02 1997−08−15 1998−04−01 1998−11−12 1999−06−30 2000−02−11 2000−09−26 2001−05−11 2001−12−31
0
01/01/2012
(c) Tech Bubble: Technology Stock Index
Source: calculated by data from CRSP
01/06/2012
31/10/2012
01/04/2013
31/08/2013
30/01/2014
01/07/2014
30/11/2014
01/05/2015
(d) Bitcoin Bubble 2013: Bitcoin Price Index
Source: blockchain.info,bitcoin.org
Figure 9: Price index paths of historical bubble events
there is perfect tax credit and there existed an N T bubble when assuming there is no tax credit in each
of the four events22 .
For all events of bubble, their parameters, as well as actual bubble sizes, inferred η and theoretical
bubbles sizes are listed in Table 1. Before we present the graphical comparison of theoretical and empirical
results, we briefly discuss some aspects of these historical events and explain the calculation in more details.
Parameters
c
Tech 0.00557
1929 0.0179
Japan 0.0355
Bitcoin 6.28
R
0.0144
0.0471
0.0730
0.0242
Data
τ
0.35
0.12
0.375
0.15
B
2.02
3.82
2.33
130
η
4.61
4.71
3.62
256
c
η
Perfect Credit
Theory Data
R
0.0012
0.00380
0.00979
0.0245
η
0.00312
0.0100
0.0201
0.0001
B
η
B
η
0.894
0.918
0.718
0.686
0.440
0.812
0.644
0.509
η
4.85
4.78
4.37
266
No Credit(Case NT)
Theory Data
c
R
η
η
0.00115
0.00374
0.00811
0.0236
0.00301
0.00985
0.0167
0.00009
Table 1: Summary of parameters and results of historical bubble events
22
It can also be verified that the conditions for the T bubble are not satisfied when assuming no credit.
27
B
η
B
η
0.604
0.727
0.518
0.606
0.417
0.800
0.534
0.491
1929 Stock Market Crash The Roaring Twenties witnessed great economic growth in the Unite States,
evidenced by the large-scale use of automobiles, telephones, motion pictures, electricity. Technology innovations and the extreme optimism undergirded the stock market appreciation. People had the illusion that
the bull market would last forever, which was broken in October 1929 and a decade depression ensued.
This illusion held by some investors is characterized by the belief dispersion in our model. We define the
run-up as the period from August 24, 1921 to September 3, 1929, with the latter being the peak. In all
historical events we study, we set the S&P index at the beginning of the run-up to unity. Hence the index
at the peak is 5.97. The mean index after the peak is 2.14 and the actual size of the bubble is therefore
measured as 5.97-2.14=3.83. The largest price drop within a period of 90 trading days started on Mar 23,
1923 with a index value of 1.64. The broker commission rates around 1929 were about 0.3%23 , which gives
c = 0.3% × 5.97 = 0.0179. The 3-month return from a 3 month treasury bond at June, 1929 was 3.16%,
which gives R =
3.16%
4
× 5.97 = 0.047. We treat the main contributors of the price run-up as corporations
and hence use the corporate tax rate 12%24 . The capital loss carry-over was first introduced in Revenue
Code 1939, which was restricted to individuals only, and we were not able to find any information about
capital loss carry-over around 1929. When assuming there was perfect tax credit, we calculate η from
Ps∗ (θ − η2 ) = 1.64, where Ps∗ (·) is from Proposition 5.1. Then we calculate the theoretical B by using
the inferred η and normalize both actual and theoretical bubble sizes by dividing them by the inferred η.
When assuming there was no tax credit, the calculation is similar except we use Proposition 6.1.
The calculations for all other events are the same as explained above.
Japan Asset Bubble in the 1980s By the end of 1989, Japan has experienced an “economic miracle”
that lasted more than three decades. Japanese industry gained its competitive advantage by improving
technologies from Western products and selling them back to the West for cheaper prices. In the 1970s
and 1980s Japanese high-quality and fuel-efficient automobiles gained significant market shares in the US
and consumer electronics products (e.g. VHS recorder and the Sony Walkman) started to dominate global
markets. Rapid accumulation of wealth by both corporations and households, as well as uncontrolled
money supply and credit expansion greatly encouraged speculative trading and inflated real estate and
stock market prices. Evidences show that corporations and investors’ opinion and expectations about
stock prices were highly differentiated25 . Although the bubble in the stock market and real estate are
23
See Jones (2002).
At that time the tax codes did not differentiate operation gains and capital gains.
25
See Shiller et al. (1996).
24
28
intertwined, the former burst a year ahead of the latter and so we choose to study the stock market due
to the potential causality. The Nikkei 225 index reached a record high on Dec 29, 1989, which was 5.07
times of the index in 1982. After that the index plummeted to half its peak in eight months and the
bursting of bubbles contributed to what many call the Lost Decade. We define the run-up starting from
January 1, 1982 to the end of 1989. For the largest price drop within a period of 180 trading days, we
ruled out the episode coinciding the 1987 stock market crash in the US and adopt the next largest one,
which started on August 20, 1986 with a index value of 2.47. We use the corporate tax rate (37.5%)26 as
the main contributors of stock market bubble were corporations27 . Unfortunately we were not able to find
information about capital loss carry-over round the late 1980s.
Tech Bubble in the late 1990s In the 1980s and 1990s computer started to become a consumer
electronics due to the dramatic price declines in hardware and the progress in software development. The
emergence of internet in the 1990s opened a door to an unlimited potential of how people could search
for and retrieve information and knowledge and communicate with each other, and the information and
computer related industries experienced sustained exponential growth during that time. As many other
technology innovations in the history, the true value of internet remained unclear and opinions about
how it can change our life and how soon it will happen differed substantially. Investors decide to buy
in anticipation of further rises and many technology companies in the United States were significantly
overvalued. At the height of the boom, it was possible for a promising dot-com to make an initial public
offering of its stock and raise a substantial amount of money even though it had never made a profit. This
bubble burst in March, 2000 and the technology sector in the United States suffered significant losses. We
define the run-up as from January 2, 1997 to March 24, 2000 and and compose a technology stock index
by using Nasdaq daily trading data in the technology sector (firms with SIC code 73728 and we restrict
ourselves to ordinary common shares29 ). This index is treated the single asset representing the technology
industry. For the largest price drop of this index during the run-up, we exclude the one at the beginning of
January 2000 (caused by the concerns of putting off capital gains tax payment) and the episodes coinciding
with the Russian sovereign debt default and the failure of Long-Term Capital Management, we use the next
largest one, which started on October 27, 1997. As argued by Griffin et al. (2011), the main contributors
26
In 1989 the corporate tax did not differentiate ordinary corporate income and capital gains in Japan.
See Stone and Ziemba (1993).
28
A SIC code starting with 737 means the firm engages in computer programming, data processing and other computerrelated services.
29
CRSP share code 10 or 11
27
29
to the Tech bubble were institutional traders. We use the short-term capital gains tax rate at the time for
the highest tax bracket, 35%, given the fact that there were (unsuccessful) attack around April 1999 and
many institutional investors may had repurchased the asset after the attack. As stated in the introduction,
corporate capital losses in the United States can be carried 3 years back and 5 years forward without an
dollar limit. This tax credit policy is somewhere in between null and perfect.
Bitcoin Bubble in 2013 Bitcoin is an online payment system released as an open-source software in
2009. Users can transact directly without any financial intermediary. The system is a peer-to-peer one,
meaning that it works without central administration and that transactions are executed and verified by
randomly chosen network nodes and recorded in a public distributed ledger called the block chain. Any
online computer installed with this software becomes a node. This feature ensures that no one can control
the bitcoin system. There are many online merchants where one can purchase bitcoin with local currency.
Once bought, the amount of holding is recorded in the ledger. Since a holder does not know where this
information is stored and all data is encrypted and verified, it is impossible for a user to tamper the ledger.
Users who install the bitcoin software and thus facilitates the distributed transactions, which is called
mining, are rewarded by newly created bitcoin, though this reward is declining as more and more nodes
are online. There is an upper bound of 21 million for the total bitcoin that can be circulated in the system.
Bitcoin as a virtual currency and a form of payment for products and services has grown and merchants
have an incentive to accept it because fees are lower than credit cards. The price for the bitcoin grew
slowly before 2012. Since 2012 its price rose considerably from around US$5 and soared in 2013, hitting
its record high at US$1151 on December 4. After that the price declined sharply and was around US$50
in May 2015. The market capitalization at the height of the bubble was close to 14 billion US dollars.
The gains from investment on virtual currency such as bitcoin in the 2013 tax year became taxable in the
United States30 . We define the run-up as between January 1, 2012 and December 4 2013, and scale the
price on January 1, 2012, which was US$5.2, to 1. The nominal transaction costs are very low, but we
need to take into account the impact of newly created bitcoin, which is similar to excess money supply that
can cause inflation. As such, we use the miners revenue as a percentage of the transaction volume for the
transaction cost. The largest price drop during the run-up within 35 trading days started on April 9, 2013.
We conjecture that the majority of investors of bitcoin are individuals, and they are tech-savvy young
people with moderate income. The tax rates applied to annul single income brackets between $8,926 and
30
See Internal Revenue Service Notice 2014-21.
30
$87,850 are 15% and 25%, but considering that personal capital gain can offset all other personal losses
(not just capital gains) and tax carry-over never expires, we choose 15% in the calculation.
In Figure 10 we depict these actual bubbles and compare them the theoretical results. Panel (10a)
assumes a perfect tax credit and tax rates are irrelevant. Therefore we can use
2c+R
η
as a single parameter
that determines the size of the bubble. In Panel (10b) we assume there is no tax credit, in which case tax
can affect the bubble. Since it is very difficult to graphically illustrate the impacts of three parameters (τ ,
in DN T (hence in B) in Proposition 6.131 . As such we can illustrate
c and R) on B, we drop the term −τ 2c
η
the theoretical bubble size as a function of two parameters, τ and
2c+R
.
η
1929
B
η
B
η
Japan
B
η
Tech
Bit Coin
Tech
τ
τ
2c+R
η
2c+R
η
(a) Assume there is perfect tax credit
(b) Assume there is no tax credit
Figure 10: Comparison: Theoretical and actual bubbles
7.2
Other Implications
Second, our model shows that it is possible to detect trader’s private beliefs from high-frequency trading
data. Suppose that we can identify the passive agent (usually long-term investors such as pension funds,
insurance companies and other institutional traders that invest conservatively belong to this class) and
rational (speculative) traders. Then when there is a perfect tax credit, or when there is no tax credit but ηc ,
τ and
31
R
η
are relatively large (T bubble), all traders with the same belief sell at the same price, which makes
When τ is smaller than one and
events we study here.
c
η
is close to zero, the impact of this drop can be ignored. Indeed this is the case in all
31
2c+R
η
selling prices perfect indicators of private belief (signal). Investors’ private beliefs are highly interesting
quantities in academic analysis, but are usually difficult to identify. Given high-frequency trading data,
they become identifiable in the above areas of the parameter space in our model. When there is no tax
credit in the N T bubble, this perfect relationship does not exist. Instead, a unique prediction is that, an
N T trader’s selling price is decreasing in her purchasing price. When we have an acceptable proxy for
private signal v, this negative relationship can be tested.
Third, the response of the margin between buying and selling prices to model parameters can be tested.
In our model, the gap between a trader’s stop-buy price and selling price is increasing in
c
η
and
R
.
η
When
there is no tax credit (or imperfect), this gap is also increasing in τ . With high-frequency trading data
where traders’ action can be tracked (but identities can still remain anonymous), the minimum and average
gaps between traders’ stop-buy price and selling price can be identified and above predictions are testable.
Lastly, our model implies that the response of price dynamics and trading volume to the changes of
certain model parameters in our model are not only empirically testable, but also consistent with existing
evidences. When there is a perfect tax credit, τ has not effect at all. An increase in c, on the other
hand, will not change selling strategies, but will shift stop-buy strategies downwards, which broadens the
gap between purchase and selling prices of all future buyer (who have not bought yet). This implies that
traders in the gray area in Figure 11 suddenly switch from buy to not-buy. Under the perfect capital inflow
p
Ps∗ (v)
v+
old Pb∗ (v) = v +
η
2
η
2
new Pb∗ (v) = v +
√
−2 cη
η
2
√
−2 c0 η
current price
v
Figure 11: Downward shift of stop-buy strategies
assumption, the increase in c will not affect the price path, because there are infinite traders on the right of
the grey area willing to buy and continue to receive loans. But in the real market there are finite investors,
and those who are willing to buy may need time to finance their trade either by selling other assets first or
by borrowing. When a group of buyers suddenly lose their appetite, demand from the remaining buyers
32
will decrease, at least temporarily. With the supply side unaffected, the price appreciation will slow down
(if the bubble has not burst), and the trading volume will decrease as well.
When there is no tax credit, rises in τ and c will push the situation away from an N T bubble and closer
to T bubble (if not already in T ), not only shifting stop-buy strategies Pb∗ (·) downwards but also shifting
some traders’ selling strategies upwards. Specifically, the rises will shift the dividing line (v −2c−DT
or N T )
upwards in both N T and T bubbles, causing some N T traders switch to T traders, so that now they sell at
higher price v + η2 . The remaining N T traders’ selling strategies, as well as their lower bound Ps∗ (v, Pb∗ ), are
all shifted up (those already in case T before the change will not alter their selling strategies). Consequently,
under realistic market conditions (capital inflow is not perfect, and the behavioral passive agent does not
adjust their supply and provide the asset quickly enough), rises in τ and c will (temporarily) decrease the
market supply and demand. This results in a lower trading volume, although the net effect on asset price
is ambiguous. If τ or c decreases, the results will be exactly the opposite.
These results are consistent with empirical evidences. That a rise in τ or c causes N T traders to sell
at higher prices and reduces supply is the lock-in effect, and that a rise in τ or c drives stop-buy strategy
downward and reduces demand is the capitalization effect in the empirical literature. Seida and Wempe
(2000) show that after 1986 Tax Reform Act (capital tax increase) individual investors were less willing to
sell accrued gain stocks. Slemrod (1982) and Hanlon and Pinder (2007) find that after the Revenue Act
of 1978 in US and Australia 1999 tax reforms, respectively (both involve capital gains tax reductions),
trading volumes significantly increased. Dai et al. (2008) demonstrates that the equilibrium impact of
capital gains taxes reflects both the capitalization effect and the lock-in effect, and both Dai et al. (2008)
and Ayers et al. (2008) documented increased supply and demand after the 1997 Taxpayer Relief Act in
US (capital gains tax reduction).
8
8.1
Policy Implications
Capital loss tax credit
Our paper shows that capital gains tax will lower speculators’ expected profit; but under perfect tax credit
their behavior does not change at all, i.e. they behave as if there is no tax. To effectively curb a bubble,
our results suggest that a tax authority should not grant overly favorable tax credits on capital losses,
because they can entirely offset the suppressive effects of the tax on investors’ incentives and leave the
bubble unaffected. One clarification about this suggestion is that it should be viewed as an ad hoc tax
33
policy instead of a long-run, regular optimal policy. For those who believe that capital gains taxes should
be calculated on the basis of long-run average (net) capital gains, the recommendation to weaken tax
credit may seem unfair because, without tax credit, investors are taxed whenever they have gains and are
left alone when they incur losses, and hence the tax is not based on long-run average gains. So what we
suggest is to apply such policy only when there is strong concern that a bubble may be in progress in a
specific market. In this paper we do not intend to find an optimal tax policy that can automatically deflate
bubbles when they arises and otherwise does not interfere the market.
To weaken the capital loss tax credit and shift it away from being perfect and closer to no credit, tax
authorities can set or tighten a dollar limit on the tax credit; set or reduce the inclusion rate of the capital
loss, which lowers the rate τct in (8) so that τct < τ ; shorten the duration within which tax credits can
be carried forward or backward. Other measures include targeting the tax at specific source of gains and
imposing different tax rates and tax credit policies to gains from different sources. Although our model
only analyzed the two extreme cases (perfect tax credit and no credit) and does not explore some of the
practical issues (e.g. expiration of credits), the above suggestions are likely to strengthen the effectiveness
of tax and help depress bubbles.
The introduction of capital gains tax on overseas investors in April 2015 will close a tax loophole in
the United Kingdom, before which these investors are exempt from this tax. Most analysts believe that
foreign investors contributed the most to the the ongoing house price upsurge in London and that this
change is targeting at them. To take the advantage of this change to its full extent, our paper suggests
that, when imposing the tax, UK tax authority should also limit the the tax credits to foreign investors.
In addition, if under tax treaties the introduction of this tax in UK exempts foreign investors from tax
liabilities in their home countries, then the tax may has little or no effect.
8.2
Transaction taxes
Our model also shows that the marginal effect of the transaction cost c in reducing the bubble is very large
when c is small, although it diminishes as c increases. Many countries have had experimented financial
transaction taxes on trades of shares, bonds, currencies and derivatives, and some countries have abolished
the tax or lower the tax rate later. As mentioned earlier, empirical evidences show that these experiments
tend to reduce trading volumes but not price volatilities. Although we use a fixed transaction cost instead
of a proportional transaction tax, our results may nevertheless rationalize the fact that some countries still
maintain financial transaction taxes at very low rates because of its high marginal effect at low rates. For
34
example, the US currently levies a 0.0034% tax on stock transactions, UK levies a rate of 0.5% tax on
share purchases, Japan has a 0.1 − 0.3% tax on stock transactions.
8.3
Return from outside investment option and zero interest rate policy
Another implication from our model is that, if the alternative investment opportunity has a higher payoff,
it will help deflate the bubble. Conversely, a deterioration of the return on alternative investment reduces
the opportunity cost of riding the bubble and hence induces a larger bubble. Consequently, governments
should be alert on bubbles arising in domestic markets not only when major overseas markets are in booms,
but also when markets like US (the alternative investment opportunities in our model) are experiencing
recessions. The potential housing bubbles that we have observed in major cities outside US are partially
the results of arbitrageurs switching their investments from US real estate markets plagued by the subprime
mortgage crisis.
Interests on savings are often viewed as outside options to other forms of investments. When central
banks set very low interest rates, this encourages households to invest on riskier assets, such as buying
a larger or a second house. Central banks often face a dilemma: low interest rates help fight recessions
and unemployment (e.g. Federal Reserve’s responses to the 2001 recession), but also contribute to the
inflation in housing prices. Although there is no agreement that Federal Reserve’s low interest rate policy
during 2001-2005 caused the housing bubble that burst in 2008, the timing of the housing price run-up
and bursting roughly coincide with the declining and rising of the federal funds rate. Our model backs the
view that low interest rates can contribute to housing bubbles.
Bank of Canada now also faces this issue. It has lowered the rates after the 2008 financial crisis and
maintained an overnight rate at 1% since 2010. This seems to coincide with the rise of potential housing
bubbles in Canada. In January 2015, the Bank surprisingly further lowered the rate to 0.75%, partially
to fight the low expected economic growth rate due to the plunge in oil price. Our model warns that, at
the already low interest rate, further lowering interest rates has a significant marginal effect of inflating
bubbles. Such interest rate policy could be especially dangerous when there is strong suspicion that a large
housing bubble already existed in Canada at this time.
8.4
Government bailout
Our model can also be used to demonstrate that the expectations for government bailouts lead to larger
bubbles. When large investors purchase risky asset in the boom, the expectations that they are “too
35
big to fail” and that the government will bail them out once they are in trouble give them incentives to
take more aggressive positions. When the post-crash trade of a break-even trader incurs a loss for sure
(Pb∗ (v) + 2c > Pb∗−1 (Ps∗ (v)) − η2 , i.e. Equation (3)), a bailout reduces the loss and the gap between pre and
post-crash payoff in the first order condition becomes smaller, with or without the tax credit. A smaller loss
encourages traders to wait and sell at higher price, and higher selling prices induce higher stop-buy prices
and hence a larger bubble emerges. When the post-crash trade of a break-even trader can be profitable
(Pb∗ (v) + 2c < Pb∗−1 (Ps∗ (v)) − η2 , i.e. Equation (2)), the expectation for bailouts does not change selling
strategies (because it is profitable in the marginal event and there is no need for bailout), but the loss will
be smaller due to the bailout (the first integral in (2)), which leads to higher stop-buy prices and a larger
bubble. Again this is true with or without tax credit. Note that the bailout has a larger inflating effect
on bubbles when there is no tax credit than there is credit, because the the bailout reduces the loss but
also crowds out the tax credit, the there will be less loss to carry after the bailout.
During the 2008 financial crisis, the U.S. Congress and government passed the Troubled Asset Relief
Program funded at $700 billion and bailed out many large banks, insurance companies and auto companies.
These bailouts have been highly controversial since day one, and one of the criticisms is that they create
moral hazard and encourage firms to take reckless risks. Our paper formalizes this criticism in a model
of bubble and demonstrates that the expectations for bailouts created from previous bailouts will clearly
encourage agents to take excess risks which fuels the next bubble.
9
Conclusion
We study the effects of taxes, especially the capital gains tax, on asset bubbles, which has not been
explored by the existing literature. Our model incorporates purchases into the framework of Abreu and
Brunnermeier (2003). In the unique equilibrium, we find that the capital gains tax has no effect on the
size of the bubble when there is a perfect capital loss tax credit. Bubble size becomes decreasing in the
capital gains tax when there is no tax credit. Therefore dealing with bubbles with the capital gains tax not
only requires imposing the tax, but also requires tightening the policies on tax credit. In contrast, a small
fixed transaction cost can also deflate the bubble considerably, with or without tax or tax credit. We also
show that a lower return from alternative investment options may induce a larger bubble and demonstrate
that further lowering an already low interest rate has a significant inflating marginal effect. The model
also has an appealing dynamic in which the stop-buy decision and selling decision spread smoothly and
36
continuously from low types to high types and high types are still buying when low type already sold. It
would be desirable to empirically verify these results.
References
Abreu, Dilip, and Markus K. Brunnermeier (2003) ‘Bubbles and crashes.’ Econometrica 71(1), 173–204
Ayers, Benjamin, Oliver Zhen Li, and John R Robinson (2008) ‘Tax-Induced Trading around the Taxpayer
Relief Act of 1997.’ The Journal of the American Taxation Association 30(1), 77–100
Brunnermeier, Markus K. (2001) Asset Pricing under Asymmetric Information: Bubbles, Crashes, Technical Analysis, and Herding OUP Catalogue (Oxford University Press)
(2008) ‘bubbles.’ In The New Palgrave Dictionary of Economics, ed. Steven N. Durlauf and Lawrence E.
Blume (Basingstoke: Palgrave Macmillan)
Brunnermeier, Markus K., and Lasse Heje Pedersen (2005) ‘Predatory trading.’ Journal of Finance
60(4), 1825–1863
Brunnermeier, Markus K., and Martin Oehmke (2012) ‘Bubbles, financial crises, and systemic risk.’ NBER
Working Papers 18398, National Bureau of Economic Research, Inc, September
Brunnermeier, Markus K., and Stefan Nagel (2004) ‘Hedge funds and the technology bubble.’ Journal of
Finance 59(5), 2013–2040
Carlin, Bruce Ian, Miguel Sousa Lobo, and S. Viswanathan (2007) ‘Episodic liquidity crises: Cooperative
and predatory trading.’ Journal of Finance 62(5), 2235–2274
Constantinides, George M (1983) ‘Capital Market Equilibrium with Personal Tax.’ Econometrica
51(3), 611–36
Dai, Zhonglan, Edward Maydew, Douglas A. Shackelford, and Harold H. Zhang (2008) ‘Capital Gains
Taxes and Asset Prices: Capitalization or Lock-in?’ Journal of Finance 63(2), 709–742
De Long, J. Bradford, Andrei Shleifer, Lawrence H. Summers, and Robert J. Waldmann (1990) ‘Positive
feedback investment strategies and destabilizing rational speculation.’ Journal of Finance 45(2), 379–95
Doblas-Madrid, Antonio (2012) ‘A robust model of bubbles with multidimensional uncertainty.’ Econometrica 80(5), 1845–1893
Griffin, John M., Jeffrey H. Harris, Tao Shu, and Selim Topaloglu (2011) ‘Who drove and burst the tech
bubble?’ Journal of Finance 66(4), 1251–1290
Hanlon, Dean, and Sean Pinder (2007) ‘An empirical investigation of whether Australian capital gains tax
reforms influence individual investor behaviour.’ Pacific-Basin Finance Journal 15(5), 481–493
Hu, Shing-yang (1998) ‘The effects of the stock transaction tax on the stock market - Experiences from
Asian markets.’ Pacific-Basin Finance Journal 6(3-4), 347–364
Jones, Charles M (2002) ‘A century of stock market liquidity and trading costs.’ Available at SSRN 313681
Kyle, Albert S (1985) ‘Continuous Auctions and Insider Trading.’ Econometrica 53(6), 1315–35
37
Lanne, Markku, and Timo Vesala (2010) ‘The effect of a transaction tax on exchange rate volatility.’
International Journal of Finance & Economics 15(2), 123–133
Li, Mei, and Frank Milne (2014) ‘The role of a large trader in a dynamic currency attack model.’ Journal
of Financial Intermediation 23(4), 590–620
Scheinkman, Jose A., and Wei Xiong (2003) ‘Overconfidence and speculative bubbles.’ Journal of Political
Economy 111(6), 1183–1219
Seida, Jim A., and William F. Wempe (2000) ‘Do capital gain tax rate increases affect individual investors’
trading decisions?’ Journal of Accounting and Economics 30(1), 33–57
Shiller, Robert J, Fumiko Kon-Ya, and Yoshiro Tsutsui (1996) ‘Why Did the Nikkei Crash? Expanding
the Scope of Expectations Data Collection.’ The Review of Economics and Statistics 78(1), 156–64
Slemrod, Joel (1982) ‘Stock Transactions Volume and the 1978 Capital Gains Tax Reduction.’ Public
Finance Review 10(1), 3–16
Stone, Douglas, and William T. Ziemba (1993) ‘Land and Stock Prices in Japan.’ Journal of Economic
Perspectives 7(3), 149–165
Temin, Peter, and Hans-Joachim Voth (2004) ‘Riding the south sea bubble.’ American Economic Review
94(5), 1654–1668
Umlauf, Steven R. (1993) ‘Transaction taxes and the behavior of the Swedish stock market.’ Journal of
Financial Economics 33(2), 227–240
Westerhoff, Frank H., and Roberto Dieci (2006) ‘The effectiveness of Keynes-Tobin transaction taxes
when heterogeneous agents can trade in different markets: A behavioral finance approach.’ Journal of
Economic Dynamics and Control 30(2), 293–322
Xiong, Wei (2013) ‘Bubbles, Crises, and Heterogeneous Beliefs.’ NBER Working Papers 18905, National
Bureau of Economic Research, Inc, March
The Impact of Capital Gain Taxes and Transaction Costs on Asset Bubbles
Appendix A
Anticipated price path p(t)
R t u(ν)
u(ν)
dν
and
p(t)
=
α(D
(t))
we
have
dν = α−1 (p(t)). Differentiating w.r.t t
r
ν=0 p(ν)
ν=0 p(ν)
dp/dt
p
on both sides, we have u(t)
= α0 (α
−1 (p)) =⇒ u(t)dt = α0 (α−1 (p)) dp. Integrating both sides w.r.t t and p,
R p(t)
R
R
p
p
respectively, we have u(t)dt + C = α0 (α−1
dp,
where
C
is
a
constant.
Let
G(p)
≡
0 (α−1 (p)) dp, then
(p))
α
R
−1
From Dr (t) =
Rt
p(t) = G ( u(t)dt + C). Combine with boundary condition p(0) = 0, we can solve for p(t).
Appendix B
Proof of Lemma 3.1
We first prove the following lemma.
Lemma B.1. lim Ps∗ (v, Pp ) = ∞, ∀Pp .
v→∞
Proof. Assume there exists P such that P > Ps∗ (v, Pp ), ∀v. For a trader with signal v > P + η2 , she knows
for sure that θ > P . Then selling at price Ps∗ (v, Pp ) < P < θ cannot be an equilibrium strategy, because
she would rather sell at price v, for instance, if the bubble has not burst, or at θ, if the it has burst. A
contradiction.
38
Lemma B.2. There exists v 0 (and Pp ) such that Ps∗ (v 0 , Pp ) = Pb∗ (v), ∀v ≥ η2 .
Proof. The lowest possible θ is zero, hence the bubble cannot burst below Pb∗ ( η2 ), and no trader should sell
below Pb∗ ( η2 ). For trader v = − η2 , she know for sure that θ = 0 and the bubble will burst at Pb∗ ( η2 ). Hence
she will sell exactly at Pb∗ ( η2 ). Because both Pb∗ (·) and Ps∗ (·, ·) are continuous, and Ps∗ (·, ·) does no have an
upper bound, the lemma holds.
Corollary 3.1 implies that Pb∗ (·) is weakly increasing. Suppose that there exists η2 ≤ v < v such that
Pb∗ (v) = Pb∗ (v). Then the probability of bursting at Pb∗ (v) is strictly positive. By Lemma B.2, there exists
v such that Ps∗ (v, Pp ) = Pb∗ (v) + ε, where ε > 0. Consider trader v. When ε is small enough, the extra
benefit, ε, from selling at Pb∗ (v) + ε instead of at Pb∗ (v) will be smaller than the loss from being caught
in the crash at Pb∗ (v). Then trader v is strictly better off by lowering her selling price from Pb∗ (v) + ε to
Pb∗ (v). A contradiction.
Appendix C
Proof of Proposition 3.1
Suppose trader v who does not hold the asset stops buying for the first time at Pb∗ (v) and then switches
back to buy. Since price is continuously and strictly rising and Pb∗ (·) is continuous, there must be trader
v + ε who does not hold the asset and has stopped buying before v could restart buying, by Assumption
3.2. By Corollary 3.1, trader v cannot restart buying before trader v + ε stops and then restarts buying,
and trader v + ε cannot do so before trader v + 2ε does so. Proceeding this way, trader v is not buying
until trader θ + η2 stops and then restarts buying. That trader θ + η2 stops buying is exactly what triggers
the burst.
If a trader v has previously sold the asset at Ps∗ (v, Pp ) and now tries to buy again, it contradicts the belief
that all lower types have stopped buying, since Pb∗ (v) ≤ Ps∗ (v, Pp ).
Appendix D
If
2c+R
η
<2−
Equilibrium strategies of trader v ≤
tax credit and positive bubble
√
3 <, a trader
η
2
−
p
(4c + 2R)η < v ≤
η
2
with perfect
η
2


buy, if price < Pb∗ (v) = HC (v);


will
sell, if price ≥ Ps∗ (v) = v + η2 ; ,



 hold, if P ∗ (v) ≤ price < P ∗ (v).
s
b
p
and 
a trader v ≤ η2 − (4c + 2R)η
p


buy, if price < η − (4c + 2R)η − 2c − R;


p
will
sell, if price ≥ η − (4c + 2R)η;



 hold, if η − p(4c + 2R)η − 2c − R ≤ price < η − p(4c + 2R)η.
√
If 2 − 3 ≤ 2c+R
< 12 , a trader v ≤ vC will never buy the asset, and a trader vC < v <
η



buy, if price < Pb∗ (v) = HC (v);


will
sell, if price ≥ Ps∗ (v) = v + η2 ;



 hold, if P ∗ (v) ≤ price < P ∗ (v).
b
s
39
η
2
v 2 +(η−2R−4c)v−2η
√
(2R+4c)η+Rη+2cη+ 5 η 2
4
where HC (v) ≡
, and vC ≡ − η2 + 2c + R+
2v+η
q
p
4c2 −η 2 −4cη+R2 +4Rc−2Rη+2η (4c+2R)η.
p
√
When 2c+R
< 2 − 3, traders with very low signals (v ≤ η2 − (4c + 2R)η) all stop buying at the same
η
p
p
price η − (4c + 2R)η − 2c − R and sell at the same price η − (4c + 2R)η, because they know that the
bubble will not burst before trader v = η2 stops buying. Hence, they all sell at Pb∗ ( η2 ) and maintain the
p
minimum gap 2c + R between their stop-buy and selling prices. For those whose v ∈ [ η2 − (4c + 2R)η, η2 ],
it is optimal for them to sell at Ps∗ (v) = v + η2 , since their purchase costs are sunk and irrelevant. However,
it is possible that the bubble bursts before they sell. In addition, since their belief support is truncated,
they
p have gaps between their stop-buy and selling prices that are larger than 2c + R but smaller than
(4c + 2R)η. √
< 12 , traders with very low signals (v ≤ vC ) never buy the asset, because their
When 2 − 3 ≤ 2c+R
η
stop-buy prices Pb∗ (v) ≤ 0.
Appendix E
Proof of Proposition 5.1
Part 1: Equilibrium verification for v > η2
Suppose that all other traders use strategies prescribed in Proposition 5.1.
√
Sale decision: Since θ = Pb∗−1 (Ps ) − η2 = Ps − θ + 2 cη, then Φ(θ|v) =
θ−(v− η2 )
η
and φ(θ|v) =
1
.
θ
d2 ω
dω
dω
= η1 (−Ps + η2 + v), and SOC is dP 2 = −1 < 0. Set dP
= 0,
Substitute into Equation (8), we have dP
s
s
s
η
η
∗
∗
∗
we have Ps (v) = v + 2 . Therefore, given all traders stop buying at Pb (v), selling at Ps (v) = v + 2 is an
equilibrium.
√
Purchase decision: Because Equation (4) is decreasing in Pb , Pb∗ (v) = v + η2 − 2 cη is uniquely pinned
down by (9). Under the strategy Pb∗ (·), B > 0 when ηc < 14 .
Part 2: Equilibrium uniqueness for v > η2
We prove the uniqueness by four lemmas. Let p∗b (v) ≡ Pb∗ (v) − v. Suppose that B > 0 upon burst.
Lemma E.1. Any equilibrium strategy must be that p∗b (v) is bounded.
Proof. The continuity of P ∗ (·) implies that P ∗ (·) is finite for finite v. Recall that P ∗ (·) is an increasing
function. If limv→∞ p∗ (v) = ∞, then ∀B > 0, there exists θ such that Pb∗ (θ + η2 ) − θ > B. Hence P ∗ (v)
cannot be an equilibrium strategy.
Lemma E.2. In any equilibrium, FOC = 0 holds for all traders in ( η2 , ∞).
∂ω
Proof. Assume in an equilibrium strategy Pb∗ (·) and Ps∗ (·), there exists a trader v such that her ∂P
6= 0.
s
∂ω
If her ∂Ps < 0, that means she can increase her expected payoff by decreasing her selling price p. The
only lower boundary for price is zero. But selling at zero is never optimal because θ > 0 for sure. Hence
for all trader v > η2 , selling at the corner solution zero cannot be optimal. Therefore in any equilibrium a
∂ω
∂ω
trader v’s ∂P
cannot be strictly negative. If her ∂P
> 0, that means she can increase her expected payoff
s
s
∂ω
by increasing her selling price p. But there is no upper boundary for price, so ∂P
> 0 itself implies this
s
strategy is not optimal. Therefore in any equilibrium a trader v’s FOC cannot be strictly positive. In
∂ω
addition, those Ps at which ω(Pp , Ps ) is non-differentiable ( ∂P
does not exist) cannot be in the equilibrium
s
∗−1
∂ω
for the following reason. ∂Ps does not exists at two points: Pb (Ps ) − η2 = v ± η2 . But Ps at neither point
can be equilibrium strategy, because Pb∗−1 (Ps ) − η2 = v + η2 =⇒ Ps = Pb∗ (v + η), which is not rational when
bubble size is strictly positive, and Pb∗−1 (Ps ) − η2 = v − η2 =⇒ Ps = Pb∗ (v), which is not rational when
transaction cost c > 0. Hence FOC= 0 holds for all traders v > η2 in equilibrium.
40
dω(P ∗ (v),P ∗ (v),v)
∂ω dPb
∂ω dPs
∂ω
s
b
= ∂P
+ ∂P
+ ∂ω
= 0. Since ∂P
= 0 in
From ω(Pb∗ (v), Ps∗ (v)) = (1 − τ )R we have
dv
∂v
s dv
s
b dv
η
η
η
∂ω
∂ω
0
0
EQ, we have ∂Pb Pb + ∂v = 0 =⇒ −Pb − (v − 2 )φ(v − 2 |v) + Ps φ(v + 2 |v) = 0. For uniform distribution,
this means
η
0
Ps∗ (v) = ηPb∗ (v) + v −
(10)
2
∂ω(Pb ,Ps )
∂Ps
= 0 (partially) and for uniform distribution, we have
η
0
0
Pb∗ (Pb∗−1 (Ps∗ (v))) − 1 =
[Pb∗ (v) − 1]
∗−1
∗
η + v − Pb (Ps (v))
Substitute (10) into FOC:
(11)
From Ps∗ (v) > Pb∗ (v) =⇒ Pb∗−1 (Ps∗ (v)) > v. From Ps∗ (v) ≤ Pb∗ (θ + η2 ) < Pb∗ (v + η2 + η2 ) = Pb∗ (v + η) =⇒
η
Pb∗−1 (Ps∗ (v)) < v + η. Hence, 0 < η + v − Pb∗−1 (Ps∗ (v)) < η =⇒ η+v−P ∗−1
> 1. Therefore, from (11)
(Ps∗ (v))
b
∗0 ∗−1 ∗
0
0
Pb (Pb (Ps (v))) > Pb∗ (v), if Pb∗ (v) > 1
we know
.
0
0
0
Pb∗ (Pb∗−1 (Ps∗ (v))) < Pb∗ (v), if Pb∗ (v) < 1
For ∀v1 , its “images” v2 , v3 , ... are defined as v2 ≡ Pb∗−1 (Ps∗ (v1 )), v3 ≡ Pb∗−1 (Ps∗ (v2 )), etc. Let xi ≡
0
0
η
η
= η+vi −v
and Xi ≡ Πij=1 xj . Then given any v1 , we have Pb∗ (vi )−1 = xi−1 [Pb∗ (vi−1 )−1] =
η+v −P ∗−1 (P ∗ (v ))
i+1
i
b
0
s
i
Xi [Pb∗ (v1 ) − 1].
Lemma E.3. limi→∞ Xi = ∞
Proof. Suppose limi→∞ Xi converges to a finite value. This implies that limi→∞ xi = 1, which in turn
implies that vi+1 → vi , i.e. Pb∗−1 (Ps∗ (vi )) → vi . Recall that Ps∗ (vi ) − Pb∗ (vi ) ≥ 2c. This means that
the slope of the line segment (vi+1 , Pb∗ (vi+1 )) − (vi , Pb∗ (vi )) can be arbitrarily large, and this slope goes
to infinity as i → ∞. Then Pb∗ (·) must diverge upward from 45◦ line, i.e. Pb∗ (v) − v is not bounded. A
contradiction.
0
Lemma E.4. Pb∗ (v) = 1, ∀v >
1
2
0
Proof. Suppose ∃v1 such that Pb∗ (v1 ) < 1. Since xi > 1, ∀i, if Xi → ∞, then there must exist vi such
0
0
0
that Pb∗ (vi ) − 1 < −1 =⇒ Pb∗ (vi ) < 0, which violates the assumption that Pb∗ (·) > 0. Suppose ∃v1 such
0
0
0
that Pb∗ (v1 ) > 1, we have limi→∞ Pb∗ (vi ) = ∞. We know that in equilibrium Ps∗ (vi ) = ηPb∗ (vi ) + vi − η2 ≤
0
0
Pb∗ (vi + η). Since Pb∗ (vi ) will grow unboundedly, we see that ηPb∗ (vi ) − 32 η ≤ Pb∗ (vi + η) − (vi + η),
which implies that Pb∗ (vi + η) − (vi + η) will also grow unboundedly, which contradicts Lemma E.1 that
p∗b (vi ) = Pb∗ (vi ) − (vi ) must be bounded.
Let Pb∗ (v) = v + g, where g is a constant. We can solve for Ps∗ (·) through standard procedure and
have Ps∗ (v) = v + η2 . Because Equation (4) is decreasing in Pb in any equilibrium with perfect tax credit,
√
√
Pb∗ (v) = v + η2 − 2 cη is uniquely pinned down by (9), i.e. g = η2 − 2 cη. Then B > 0 requires that ηc < 14 .
The belief that B ≤ 0 will lead to an entirely different strategy profile (Proposition 5.2), which cannot be
equilibrium strategy when ηc < 14 .
Part 3: Equilibrium verification and uniqueness for v ≤ η2
√
√
It is straightforward to verify the equilibrium strategies when 2 − 3 ≤ 2c+R
< 12 and when 2c+R
< 2 − 3,
η
η
so we skip the procedure.
Uniqueness: The equilibrium strategy of a trader v ∈ [− η2 , η2 ] is unique because her decision depends
only on the strategies of those whose v > η2 , which have already been pinned down in Part 2. Specifically,
for trader v ∈ [− η2 , η2 ], we can also calculate her expected profit from (7). The difference is that the lower
bound of θ is zero instead of v − η2 , and her belief about θ is a uniform distribution over [0, v + η2 ]. The
derived selling strategy is still Ps∗ (v) = v + η2 , with the restriction that Ps∗ (v) ≥ Pb∗ ( η2 ) since bubble will
√
never burst blow Pb∗ ( η2 ). Pb∗ ( η2 ) = η − 2 cη is pinned down by Ps∗ ( η2 ) from (9). Hence Ps∗ (v) equals v + η2
√
√
√
for trader v ∈ [ η2 − 2 cη, η2 ] and equals Pb∗ ( η2 ) = η − 2 cη for trader v ∈ [− η2 , η2 − 2 cη]. A trader in the
latter range will sell before crash without risk, and she will not sell strictly above Pb∗ ( η2 ) because it is not
optimal marginally. Pb∗ (v) = H(v) can also be derived from (9) for a trader v ∈ [− η2 , η2 ).
41
Appendix F
Proof of Proposition 5.2
We first verify that this is an equilibrium. When everyone else stops buying at price v − 2c − R, bubble
bursts at pT = θ + η2 − 2c − R, which is smaller than θ when 2c+R
≥ 21 . As the bubble size is non-positive,
η
it is optimal for traders to sell after the burst, i.e. they will try selling as late as possible. For trader v,
the largest possible θ is v + η2 , so selling at any price ≥ v + η2 is justified and is equivalent. We assume
that in this case traders sell at v + η2 . Then everyone simply gets the after burst price θ, which means
their expected sale revenue is E[θ|v]. Then the best response in purchase stage is to stop buying at
Pb∗ (v) = E[θ|v] − 2c − R = v − 2c − R. Then for traders with v ≤ 2c + R, their stop-buy price is zero or
negative, so they will never buy the asset.
Uniqueness: If B < 0, then selling below Pb∗ (v + η2 + η2 ) = v + η + g is strictly dominated by at or
above v + η + g, and the latter means that the trader will get θ for sure. Knowing that the expected sale
price is E[θ|v] = v, a trader will stop buying at v − 2c − R to maintain a non-negative expected payoff.
≥ 12 . The belief that B > 0 will lead to an entirely different strategy profile
Then B ≤ 0 requires that 2c+R
η
(Proposition 5.1), which cannot be equilibrium strategy when 2c+R
> 12 .
η
Appendix G
Equilibrium strategies of trader v ≤
credit and with positive bubble
η
2
√
2R+4c
without tax
η
N oT
1. When τ < 21 − (1− 2c η)2 −4 R and Rη < 18 (1 − 2c
)2 − ηc and ηc < 3−22 2 , a trader vN
T < v ≤ 2 will
η
η
η


s




v+η+DN T −2c− τ1 (2v+ 32 η+DN T )+ 2τ12 (2v+η−DN

T)





Ps L

buy, if price < Pb∗ (v) =

, if v > vN

T ;






 η + D , if v < v Ps L ;

NT
NT
2

,
η−(4c+2η+2DN T )τ
τ
1−τ



v
−
P
+
,
if
P
>
v
−
2c
−
D
;
p
NT

1−2τ
1−2τ p
2(1−2τ )

sell, if price ≥ Ps∗ (v, Pp ) =



 v + η , if P < v − 2c − D ;

p
NT

2



 hold, if P ∗ (v) ≤ price < P ∗ (v).
s
b

 v + η + D − 1 ( η + v) + DNs T , if v > v Ps L
NT
NT
t 2
2t(1−2t)
∗
∗
and in particular Ps (v, Pb (v)) =
 η + D , if v < v Ps L
NT
NT
2


buy, if price < Pb∗ (v) = η2 + DN T − 2c − R;




 sell, if price ≥ P ∗ (v, P ) = 1−τ v − τ P
p
s
1−2τ
1−2τ p
Ps H
N oT
and a trader vN T < v ≤ vN T will
η−(4c+2η+2D

N T )τ

+
;

2(1−2τ )



 hold, if P ∗ (v) ≤ price < P ∗ (v, P ).
b
ητ +2DN T −2Rτ
Ps H
Ps L
τ
N oT
where vN
,
T = DN T + 2c 1−τ , vN T = DN T + 2c, vN T =
2(1−τ )
p
s
DN T (v) ≡ (1−τ )(1−2τ )(2v+η) [η+2v−(3η+2v+4DN T −4Rτ )τ ].
42
p
s
η
and in particular Ps∗ (v, Pb∗ (v))
 = 2 + DN T


buy, if price < η2 + DN T − 2c − R;


Ps H
and a trader v ≤ vN
sell, if price ≥ η2 + DN T ;
T will



 hold, if η + D − 2c − R ≤ price <
NT
2
η
2
+ DN T .
2( 2c + R )
2. When 12 − (1− 2cη )2 −η 4R < τ <
η
and 18 (1 −
2c 2
)
η
(a) When
c
η
η
−
c
η
R
η
<
<1−
(1− 2c
)2 −2 R
η
η
< 12 (1 −
√
3
2
1− 4c
− 2R
η
η
and
a trader vTDiv < v ≤
η
2
4c
)
η
and
and
c
η
<
√
3−2 2
,
2
or when τ <
1− 4c
− 2R
η
η
(1− 2c
)2 −2 R
η
η
2
R
1−8 ηc +4 c 2 + R2 +4 Rc
2 −4 η
2 ηc
η
η
η
2
2
1−8 ηc +20 c 2 +2 R2 +12 Rc
−4 R
η
η
η
η2
<2− 3−
and τ <



buy, if price < HTT Pb (v);


will
sell, if price ≥ Ps∗ (v, Pp ) = v + η2 ; ,



 hold, if P ∗ (v) ≤ price < P ∗ (v).
,
s
b
vTN oT
c
η
< 14 ,
2
√
R
η
< 18 (1− 2c
)2 − ηc and
η
R
η
< v < vTDiv
and atrader

 H N T Pb (v), if v > v Ps L ;



T
T
∗

buy, if price < Pb (v) =
;


 η + D − 2c − R, if v < v Ps L ;


T
T
(2
will
η−(4c+2η+2DT )τ
1−τ
τ
v−
P +
if Pp > v−2c−DT ;

2(1−2τ )

sell, if price ≥ Ps∗ (v, Pp ) = 1−2τ η 1−2τ p



v + 2 if Pp < v − 2c − DT ;



∗
∗
hold, if Pb (v) ≤ price
 < Ps (v).
 v + η + D − 1 ( η + v) + DTs , if v > v Ps L ;
T
T
τ 2
2τ (1−2τ )
,
particular Ps∗ (v, Pb∗ (v)) =
η
P
L
 + D , if v < v s ;
T
2
vTPs H
and in
T
v ≤ vTN oT
and 
a trader
<


buy, if price < Pb∗ (v) = η2 + DT − 2c − R;


1
will
sell, if price ≥ Ps∗ (v, Pp ) = 1−2τ
v + η2 −τ (2c+η+v+DT +Pp ) ; , and in particular Ps∗ (v, Pb∗ (v))



 hold, if P ∗ (v) ≤ price < P ∗ (v, P ).
b
η
2
s
p
+ DT ,



buy, if price < η2 + DT − 2c − R;


and a trader v ≤ vTPs H will
sell, if price ≥ η2 + DT ;



 hold, if η +D −2c ≤ price < η +D .
T
T
2
2

T Pb


 buy, if price < HT (v);

(b) Otherwise, a trader vTPb < v ≤ η2 will
,
sell, if price ≥ η2 ;



 hold, if H T Pb (v) ≤ price < η .
T
2
a trader v < vTPb will never buy the asset.
τ
T −2Rτ
where vTPs H ≡ DT + 2c 1−τ
, vTN oT ≡ DT + 2c, vTPs L ≡ ητ +2D
, vTDiv ≡ DT − (1 − τ )R+
2(1−τ )
q
(1 − τ ) (1 − τ )R2 + (2DT + η)R − 12 (2DT + η)2 ,
p
DqTs (v) ≡ (1−τ )(1−2τ )(2v+η) [η+2v−(3η+2v+4DT −4Rτ )τ ], HTT Pb (v) ≡ v − 2c + η2 +
1−τ
1
(−2(c+DT )η+4Rv+4c(c+DT ))τ 2+(− 21 η 2+(R+4c−2v+2DT )η−2v(R+v))τ+41 (η+2v)2 ],
τ
1−2τ [
HTN T Pb (v) ≡ v + η + DT − 2c − τ1 (2v + 32 η + DT ) + 2τ12 (2v + η − DTs ), vTPb ≡ 2c − η2 + R+
r
[−(R+c+DT )η+2c2 +(4R+2DT )c+R2 ]τ 2 +[(2R+3c+2DT )η−6c2 −(6R+2DT )c− 32 R2 ]τ −(R+2c+DT )η+ 12 (R+2c)2
.
(1−τ )(1−2τ )
43
v−η
2τ
+
Appendix H
Proof of Proposition 6.1
Part 1: Equilibrium verification for v >
We can verify that Pb + 2c > Pb∗−1 (Ps ) −
Pb∗−1 (Ps )− η2 when
c
η
<
1
2
and 21 −
4c
η
(1− 2c
)2
η
η
2
η
2
when
<τ <
c
η
<
1− 4c
η
(1− 2c
)2
η
3
2
−
√
2 and τ <
1
2
−
4c
η
(1− 2c
)2
η
, and that Pb + 2c <
. Hence the first part of Proposition 6.1 corresponds
to the Equation (6) and the second part corresponds to Equation (5), both with I = 0. Following essentially
the same procedure in Part 1 of Appendix E, applied to Equation (6) and (5) respectively (both with I = 0),
we can verify that Ps∗ (·) and Pb∗ (·) define the equilibria.
Part 2: Equilibrium uniqueness for v > η2
The proofs for both case T and N T are essentially the same as Part 2 in Appendix E, so we skip them.
Part 3: Equilibrium verification and uniqueness for v ≤ η2
The proofs for both case T and N T are essentially the same as Part 3 in Appendix E, so we skip them.
Appendix I
Proof of Proposition 6.2
Equilibrium verification is essentially the same as in Appendix
F, except that the stop-buy price Pb is
R Pb +2c
R v+ η
2
pinned down by θ=v− η (θ − Pb − 2c)φ(θ|v)dθ + (1 − τ ) θ=Pb +2c (θ − Pb − 2c)φ(θ|v)dθ = (1 − τ )R and
2
q
we have Pb∗ (v) = v + η2 − τη − 2c ± τη (1 − τ )(1 − 2τ Rη ). Since Pb∗ (v) + 2c ≥ v − η2 , we have Pb∗ (v) =
q
v + η2 − τη − 2c + τη (1 − τ )(1 − 2τ Rη ). Furthermore Pb∗ (v) + 2c ≥ v − η2 requires that 2 Rη ≤ 1. The proof
of uniqueness is also the same as Appendix F.
44