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Wall Street and Silicon Valley:
A Delicate Interaction
George-Marios Angeletos
Guido Lorenzoni
MIT and NBER
MIT, Federal Reserve Bank of Chicago
NBER
Alessandro Pavan
Northwestern University
September 23, 2007
Abstract
Financial markets look at data on aggregate investment for clues about underlying pro…tability. At the same time, …rms’investment depends on expected equity prices. This generates a
two-way feedback between …nancial market prices and investment. In this paper we study the
positive and normative implications of this interaction during episodes of intense technological
change, when information about new investment opportunities is highly dispersed. Because
high aggregate investment is “good news” for pro…tability, asset prices increase with aggregate
investment. Because …rms’ incentives to invest in turn increase with asset prices, an endogenous complementarity emerges in investment decisions— a complementarity that is due purely
to informational reasons. We show that this complementarity dampens the impact of fundamentals (shifts in underlying pro…tability) and ampli…es the impact of noise (correlated errors
in individual assessments of pro…tability). We next show that these e¤ects are symptoms of ine¢ ciency: equilibrium investment reacts too little to fundamentals and too much to noise. We
…nally discuss policies that improve e¢ ciency without requiring any informational advantage on
the government’s side.
Keywords: heterogeneous information, complementarity, volatility, ine¢ ciency, beauty contests.
We thank Olivier Blanchard, Hyun Song Shin, Rob Townsend, Jaume Ventura, Iván Werning and seminar
participants at MIT, the Federal Reserve Board, the 2007 IESE Conference on Complementarities and Information
(Barcelona), the 2007 Minnesota Workshop in Macroeconomic Theory, and the 2007 NBER Summer Institute for
useful comments. Angeletos and Pavan thank the NSF for …nancial support. Email addresses: angelet@mit.edu;
glorenzo@mit.edu; alepavan@northwestern.edu.
1
1
Introduction
Financial markets follow closely the release of macroeconomic and sectoral data, looking for signals
about underlying economic fundamentals. In particular, high current levels of activity tend to
forecast high future profitability, leading to an increase in asset prices. At the same time, financial
prices affect the real economy by changing the incentive to invest for individual firms. For a startup company, higher asset prices raise the value of a potential IPO and facilitate financing from
venture capitalists. For firms already quoted on the stock market, higher asset prices raise the
value of equity issues and the market valuation of further investments within the firm.
Both directions of causation—from real activity to financial prices and from financial prices
to real investment—have been widely explored in existing theoretical and empirical work. The
literature on the impact of macroeconomic news on asset prices goes back to Chen, Roll and Ross
(1986) and Cutler, Poterba, and Summers (1989); the literature on the impact of asset prices on
real investment goes back to Brainard and Tobin (1968), Tobin (1969), Bosworth (1975), Abel and
Blanchard (1986), Barro (1990), and Morck, Shleifer and Vishny (1990).
In this paper, we document novel positive and normative implications stemming from the interaction of these two channels when agents do not share the same information. We first show
that this interaction generates a feedback mechanism between the real and the financial sector of
the economy: high investment drives up aggregate activity; financial markets interpret this as a
positive signal about future profitability; asset prices increase; this adds fuel to the initial increase
in investment. We next show that this mechanism can exacerbate non-fundamental movements
in real investment and asset prices, and can distort allocative efficiency. This mechanism seems
particularly relevant in periods of intense technological change, when information regarding the
viability and profitability of new technologies is widely dispersed across the economy.
Preview. We conduct our exercise within a neoclassical economy in which allocations would be
first-best efficient if all agents had the same information. A large number of “entrepreneurs” gets
the option to invest in a new technology. They have dispersed information about the profitability
of this technology and may sell their capital in a competitive financial market before uncertainty is
realized. The “traders” who participate in the financial market are also imperfectly informed, but
they observe aggregate investment, which provides a summary statistic of the information dispersed
among the entrepreneurs. In this environment, movements in real investment and asset prices are
driven by two types of shocks: “fundamental shocks,” reflecting actual changes in the long-run
profitability of investment, and “expectational shocks,” reflecting correlated mistakes in individual
assessments of this profitability.
The positive contribution of the paper is to study how the interaction between real and financial
activity affects the transmission of these shocks in equilibrium. Because high aggregate investment
1
is “good news” for profitability, asset prices increase with aggregate investment. As a result, an
endogenous complementarity emerges in investment decisions. An entrepreneur anticipates that the
price at which he might sell his capital will be higher the higher the aggregate level of investment.
He is thus more willing to invest when he expects others to invest more. In equilibrium, this
complementarity induces entrepreneurs to rely more on common sources of information regarding
profitability, and less on idiosyncratic sources of information. This is because common sources of
information are relatively better predictors of other entrepreneurs’ investment choices, and hence
of future financial prices. For the same reason, the entrepreneurs’ choices become more anchored
to the common prior, and hence less sensitive to changes in the underlying fundamentals. It follows
that the feedback between the real and the financial sector of the economy amplifies the impact of
common expectational shocks while also dampening the impact of fundamental shocks.
The normative contribution of the paper is to study whether the reaction of the economy to
different shocks is optimal from a social perspective. The mere fact that entrepreneurs care about
the financial market’s valuation of their investment does not, on its own, imply any inefficiency.
Indeed, as long as all agents share the same information, equilibrium asset prices coincide with the
common expectation of profitability; whether entrepreneurs try to forecast fundamentals or asset
prices is then completely irrelevant for efficiency.
This is not the case, however, when information is dispersed. The sensitivity of asset prices
to aggregate investment induces a wedge between private and social returns to investment: while
the fundamental valuation of the investment made by a given entrepreneur is independent of the
investments made by other entrepreneurs (i.e., there are no production externalities or spillovers),
the market valuation is not. By implication, the complementarity that emerges in equilibrium due
to the dispersion of information is not warranted from a social perspective. It then follows that the
positive effects documented above are also symptoms of inefficiency: equilibrium investment reacts
too little to fundamental shocks and too much to expectational shocks.
We conclude by examining policies that improve efficiency without requiring the government
to have any informational advantage vis-a-vis the market. We first consider interventions “during
the fact,” while information remains dispersed. In particular, we consider a tax on financial trades
or other policies aimed at stabilizing asset prices. By moderating the reaction of asset prices to
aggregate investment, these policies dampen the equilibrium impact of non-fundamental shocks,
which improves efficiency. In so doing, however, these policies also dampen the equilibrium impact
of fundamental shocks, which was inefficiently low to start with. It follows that these policies can
raise welfare, but never achieve full efficiency.
We next consider interventions “after the fact,” when uncertainty has been resolved. Building
on results from Angeletos and Pavan (2007b), we show that full efficiency can be achieved by
2
introducing a tax on capital holdings that is contingent on both realized aggregate investment and
realized profitability. Although real and financial decisions are sunk by the time these taxes are
collected, the anticipation of these contingencies affects the incentives entrepreneurs and traders
face “during the fact.” By appropriately designing these contingencies, the government can induce
agents to respond efficiently to different sources of information, even if it can not directly monitor
these sources of information.
Discussion. The US experience in the second half of the 90’s has renewed interest in investment
and asset-price booms driven by apparent euphoria regarding new technologies (e.g., the Internet),
and on the optimal policy response to these episodes. A common view in policy discussions is
that entrepreneurs and corporate managers are driven by noise traders and other irrational forces
in financial markets, or are irrational themselves. Elements of this view are formalized in Shiller
(2000), Cecchetti et al (2000), Bernanke and Gertler (2001), and Dupor (2005). Similar concerns
are currently raised for the investment boom in China. The presumption that the government can
detect “irrational exuberance” then leads to the result sought—that it should intervene.
While we share the view that expectational errors may play an important role in these episodes,
we also recognize that these errors may originate from noise in information rather than irrationality.
Furthermore, we doubt the government’s ability to assess fundamentals better than the market as
a whole. Our approach is thus different. On the normative side, we identify an informational
externality that can justify intervention even by a policy maker with no superior information.
At the same time, on the positive side, we show that the interaction between real and financial
activity can amplify the impact of noise and that this amplification is stronger when information is
more dispersed. This helps explain, without any departure from rationality, why periods of intense
technological change, like the 90’s, may feature significant non-fundamental volatility.
Because the source of both amplification and inefficiency in our model rests on the property that
investment is largely driven by expectations about others’ choices rather than about fundamentals,
our results are reminiscent of Keynes’ famous beauty-contest metaphor:
“...professional investment may be likened to those newspaper competitions in which
the competitors have to pick out the six prettiest faces from a hundred photographs,
the prize being awarded to the competitor whose choice most nearly corresponds to the
average preferences of the competitors as a whole; so that each competitor has to pick,
not those faces which he himself finds prettiest, but those which he thinks likeliest to
catch the fancy of the other competitors...” Keynes (1936, p.156).
Implicit in Keynes’ argument appears to be a normative judgement that something goes wrong
when investment is driven by higher-order expectations. However, Keynes does not explain why
3
this might be the case. More recently, Allen, Morris and Shin (2005), Bacchetta and Wincoop
(2005) and Cespa and Vives (2007) have revisited rational-expectations models of asset pricing and
have shown that, at least in certain cases, a mechanism similar to the one articulated by Keynes
increases the impact of the common prior and of common noise on equilibrium prices. However,
these papers abstract from the real sector of the economy. They also do not address whether the
positive effects they document are symptoms of allocative inefficiency. To the best of our knowledge,
our paper is the first to provide a complete micro-foundation for beauty-contest-like inefficiencies
in the interaction between real and financial activity.
Other related literature. By focusing on the two-way feedback between real and financial
decisions as a potential explanation of “bubbly” episodes, the paper also relates to two other lines of
work. One line studies rational bubbles in economies with financial frictions or asset shortages (e.g.,
Ventura, 2003; Caballero, 2006; Caballero, Farhi and Hammour, 2006). The mechanisms studied
in these papers also generate significant non-fundamental movements, but they are unrelated to
information. The second and more closely related line studies speculative fluctuations in prices and
investment due to heterogeneous priors regarding profitability (e.g., Scheinkman and Xiong, 2003;
Gilchrist, Himmelberg, and Huberman, 2005; Panageas, 2005). In these papers, investment and
prices are largely driven by expectational shocks regarding others’ valuations. This is similar to the
role of higher-order expectations in our paper. However, in these papers asset prices continue to
reflect the social value of investment, ensuring that no inefficiency emerges.1 In our paper, instead,
the impact of higher-order expectations is also the source of inefficiency.
The paper also relates to the growing macroeconomic literature on heterogeneous information and strategic complementarities (e.g., Amato and Shin, 2006; Angeletos and Pavan, 2007a,b;
Baeriswyl and Conrand, 2007; Hellwig, 2005; Hellwig and Veldkamp, 2007; Lorenzoni, 2006, 2007;
Mackowiak and Wiederholt, 2006; Woodford, 2002). However, unlike the complementarities considered in these papers, which originate in monopolistic price competition, production or demand
spillovers, or other payoff externalities, the complementarity documented here is due to an informational externality: it emerges only when information is dispersed and only because aggregate
activity is then a signal of the underlying fundamentals. This specific source of complementarity is
the key to both the positive and the normative results of our paper.2
Also related are Subrahmanyam and Titman (2001), Goldstein and Guembel (2003), and Ozde1
In Gilchrist, Himmelberg, and Huberman (2005), inefficiency can arise due to the monopoly power of the owners
of the firms issuing speculative stocks.
2
Morris and Shin (2002) show that more precise public information can have a detrimental effect on equilibrium
welfare in a game where the complementarity in individual actions reflects a discrepancy between private and social
objectives; but they do not study the origin of this discrepancy and they assume it to be exogenous to the information
structure. In our environment, instead, the discrepancy between private and social objectives, like the complementarity, originates in the two-way feedback between real and financial decisions and crucially depends on the information
structure; this has also interesting implications for the social value of information (see Section 5.3).
4
noren and Yuan (2007). These papers study a variety of feedback effects between a financial market
and a real sector that features of a form of network externality. Like the papers mentioned in the
previous paragraph, the complementarity in these papers is exogenous to the information structure.
On the other hand, a complementarity that originates in an informational externality is featured
in the currency-crises model of Goldstein, Ozdenoren and Yuan (2007); but it is of a different kind
than ours. In their model, the central bank looks at the size of attack to learn about the underlying
fundamentals. The larger the attack, the worse the bank’s perception of the fundamentals. But
then also the higher the bank’s willingness to abandon the peg and hence the higher the incentive
for the individual speculator to attack.
Layout. Section 2 introduces the baseline model. Section 3 characterizes the equilibrium and
derives the positive implications of the model. Section 4 characterizes the socially efficient use of
information and contrasts it to the equilibrium. Section 5 discusses policy implications. Section 6
considers a number of extensions. Section 7 concludes. All proofs are in the Appendix.
2
The baseline model
We consider an environment in which heterogeneously informed agents choose how much to invest
in a “new technology” with uncertain returns. After investment has taken place, but before uncertainty is resolved, agents trade financial claims on the returns of the installed capital. At this
point, the observation of aggregate investment partially reveals the information that was dispersed
in the population during the investment stage.
Timing, actions, and information. There are four periods, t ∈ {0, 1, 2, 3}, and two types
of agents: “entrepreneurs,”who first get the option to invest in the new technology, and “traders,”
who can subsequently purchase claims on the installed capital of the entrepreneurs. Each type is
of measure 1/2; we index entrepreneurs by i ∈ [0, 1/2] and traders by i ∈ (1/2, 1].
At t = 0, nature draws a random variable θ from a Normal distribution with mean µ > 0 and
variance 1/πθ (i.e., πθ is the precision of the prior). This random variable represents the exogenous
productivity of the new technology and is unknown to all agents.
At t = 1, the “real sector” of the economy operates: each entrepreneur decides how much to
invest in the new technology. Let ki denote the investment of entrepreneur i. The cost of this
investment is ki2 /2 and is incurred within the period. When choosing investment, entrepreneurs
have access to various signals (sources of information) that are not directly available to the traders.
Some of these signals may have mostly idiosyncratic noise, while others may have mostly common
noise (correlated errors). To simplify, we assume that entrepreneurs observe two signals. The
first one has only idiosyncratic noise and is given by xi = θ + ξi , where ξi is Gaussian noise,
5
independently and identically distributed across agents, independent of θ, with variance 1/πx (i.e.,
πx is the precision of the idiosyncratic signal). The second has only common noise and is given by
y = θ + ε, where ε is Gaussian noise, common across agents, independent of θ and of {ξi }i∈[0,1/2] ,
with variance 1/πy (i.e., πy is the precision of the common signal). The more general case where
all signals have both idiosyncratic and common errors is examined in the Supplementary Material.
At t = 2, the “financial market” opens: some entrepreneurs sell their installed capital to the
traders. In particular, we assume that each entrepreneur is hit by a “liquidity shock” with probability λ ∈ (0, 1). Liquidity shocks are i.i.d. across agents, so λ is also the fraction of entrepreneurs
hit by the shock. Entrepreneurs hit by the shock are forced to sell all their capital to the traders.
For simplicity, entrepreneurs not hit by the shock are not allowed to trade any claims on installed
capital.3 The financial market is competitive and p denotes the price of one unit of installed capital. When the traders meet the entrepreneurs hit by liquidity shocks in the financial market, they
R1
observe the aggregate level of investment from period 1, K = 0 ki di. They can then use this
observation to update their beliefs about θ.4
Finally, at t = 3, θ is publicly revealed, each unit of capital gives a cash flow of θ to its owner,
and this cash flow is consumed.
Payoffs. All agents are risk neutral and the discount rate is zero. Payoffs are thus given by ui =
ci1 + ci2 + ci3 , where cit denotes agent i’s consumption in period t. First, consider an entrepreneur.
If he is not hit by the liquidity shock his consumption stream is (ci1 , ci2 , ci3 ) = (−ki2 /2, 0, θki ), so
that his payoff is ui = −ki2 /2 + θki . If he is hit by the shock, he sells all his capital at the price p
and his consumption stream is (ci1 , ci2 , ci3 ) = (−ki2 /2, pki , 0), so that his payoff is ui = −ki2 /2 + pki .
Next, consider a trader and let qi denote the units of installed capital he purchases in period 2. His
consumption stream is (ci1 , ci2 , ci3 ) = (0, −pqi , θqi ), so that his payoff is ui = (θ − p)qi .
Remarks. The two essential ingredients of the model are the following: (i) the agents who
make the initial investment decisions have dispersed private information, so that aggregate investment is a signal of the fundamental; (ii) there is some common source of “noise” that prevents
aggregate investment from perfectly revealing the fundamental to all agents, so that the dispersion
of information does not completely vanish by the time agents meet in the financial market.
The specific information structure we have assumed is a convenient way to capture these two
properties. In particular, the role of the common signal y is to introduce correlated errors in the
entrepreneurs’ assessments of profitability in stage 1, thereby adding noise to the inference problem
that the traders face in stage 2: in equilibrium, aggregate investment will move both with the
3
We relax this assumption in Section 6.
Letting the traders observe the entire cross-sectional distribution of investments does not affect the results. This
is because, in equilibrium, this distribution is Normal with known variance; it then follows that the mean investment
contains as much information as the entire cross-sectional distribution.
4
6
fundamental θ and with the common error ε, ensuring that aggregate investment reveals θ only
imperfectly. As mentioned above, in the Supplementary Material we dispense with the common
signal y and instead consider the case where entrepreneurs observe multiple private signals, all
of which have both idiosyncratic and common errors. We also consider a variant that introduces
unobserved common shocks to the entrepreneurs’ cost of investment as an alternative source of
noise in aggregate investment. In both cases, our main positive and normative results (Corollaries
1 and 2) remain intact, highlighting that the key for our results is the existence of a common source
of noise, not the specific form of it.
A similar remark applies to other simplifying modeling choices. For example, we could have
allowed the entrepreneurs that are not hit by a liquidity shock to participate in the financial market;
we could further have allowed all entrepreneurs to observe a noisy signal of aggregate investment,
or a noisy price signal, at the time they make their investment decisions.5 What is essential for
our results is only that the dispersion of information remains present both at the investment and
at the trading stage.
Also note the “liquidity shock” need not be taken too literally. Its presence captures the more
general idea that when an agent makes an investment decision, be him a start-up entrepreneur or
the manager of a public company, he cares about the market valuation of his investment at some
point in the life of the project. A start-up entrepreneur may worry about the price at which he will
be able to do a future IPO; a corporate manager may be concerned about the price at which the
company will be able to issue new shares. In what follows, we interpret λ broadly as a measure of
the sensitivity of the firms’ investment decisions to forecasts of future equity prices.6
Finally, note that there are no production spillovers and no direct payoff externalities of any
kind: both the initial cost (−ki2 /2) and the eventual return on capital (θki ) are independent of
the investment decisions of other agents. The strategic complementarity that will be identified in
Section 3.1 originates purely in an informational externality.
A benchmark with no informational frictions. Before we proceed, it is useful to examine
what happens when the dispersion of information vanishes at the time of trading in the financial
market. That is, suppose that all the information that is dispersed during period 1 (namely, the
signals {xi }i∈[0,1/2] and y) becomes commonly known in period 2. The fundamental θ then also
becomes commonly known and the financial market clears if and only if p = θ. It follows that the
expected payoff of entrepreneur i in period 1 reduces to E[ui |xi , y] = E[θ|xi , y]ki − ki2 /2, which in
5
See Section 6 for these extensions.
See Baker, Stein and Wurgler (2003) for complementary evidence that the sensitivity of corporate investment to
stock prices is higher in sectors with tighter financing constraints (which here can be interpreted as higher λ).
6
7
turn implies that equilibrium investment is given by
ki = E[θ|xi , y] =
πy
πθ
πx
µ+
xi +
y.
πθ + πx + πy
πθ + πx + πy
πθ + πx + πy
The key result here is that equilibrium investment is driven solely by first-order expectations
regarding the fundamental and is independent of the intensity of the entrepreneurs’ concern about
financial prices (as measured by λ). This result does not require θ to be perfectly known in period
2. Rather, it applies more generally as long as the asymmetry of information about θ vanishes in
period 2.7 This case, which we henceforth refer to as the case with “no informational frictions,”
provides a convenient reference point for the rest of the analysis.
3
Equilibrium
Individual investment is described by a function k : R2 → R so that k(x, y) denotes the investment
made by an entrepreneur with information (x, y). Aggregate investment is then a function of (θ, y):
K(θ, y) =
Z
k (x, y) dΦ(x|θ),
(1)
where Φ(x|θ) denotes the cumulative distribution function of x given θ. Since traders observe
aggregate investment and are risk neutral, the unique market-clearing price is p = E[θ|K], where
the latter denotes the expectation of θ given the observed level of K.8 Since K is determined by
(θ, y), p is also a function of (θ, y). We thus define an equilibrium as follows.
Definition 1 A (symmetric) equilibrium is an investment strategy k(x, y) and a price function
p(θ, y) that satisfy the following conditions:
(i) for all (x, y),
k (x, y) ∈ arg max E
k
(1 − λ) θk + λp (θ, y) k − k2 /2 | x, y ;
(ii) for all (θ, y),
p (θ, y) = E [ θ | K(θ, y) ] ,
where K(θ, y) =
R
k (x, y) dΦ(x|θ).
7
To clarify this point, consider an arbitrary information structure. Let Ii,t denote the information of agent i
in period t. Impose that no agent has private information about θ in period 2 so that E[θ|Ii,2 ] = E[θ|I2 ] for all
i. From market clearing we then have that p = E[θ|I2 ]. From the law of iterated expectations we then have that
E[p|Ii,1 ] = E[E[θ|I2 ]|Ii,1 ] = E[θ|Ii,1 ] for all i. It follows that every entrepreneur chooses ki = E[θ|Ii,1 ].
8
Since the price is only a function of K and K is publicly observed, the price itself does not reveal any additional
information. Therefore, we can omit conditioning on p. The case where p conveys additional information is examined
in Section 6.
8
Condition (i) requires that the entrepreneurs’ investment strategy be individually rational,
taking as given the equilibrium price function. Condition (ii) requires that the equilibrium price
be consistent with rational expectations and individual rationality on the traders’ side, taking as
given the strategy of the entrepreneurs.
As it is often the case in the literature, tractability requires that we restrict attention to equilibria in which the price function is linear.
Definition 2 A linear equilibrium is an equilibrium in which p(θ, y) is linear in (θ, y) .
3.1
Endogenous complementarity
The optimality condition for the entrepreneurs’ strategy can be written as
k (x, y) = E [ (1 − λ) θ + λp (θ, y) | x, y ] .
(2)
The linearity of p (θ, y) in (θ, y) and of E [θ|x, y] in (x, y) then guarantees that the entrepreneurs’
strategy is linear in (x, y); that is, there are coefficients (β0 , β1 , β2 ) such that
k (x, y) = β0 + β1 x + β2 y.
(3)
By implication, aggregate investment is given by K = β0 + β1 θ + β2 y = β0 + (β1 + β2 )θ + β2 ε.
Observing K is thus informationally equivalent to observing a Gaussian signal z with precision πz ,
where
K − β0
β2
z≡
=θ+
ε
β1 + β2
β1 + β2
and
πz ≡
β1 + β2
β2
2
πy .
(4)
Standard Gaussian updating then gives the expectation of θ given K as a weighted average of the
prior and the signal z:
E [θ|K] =
πθ
πz
µ+
z.
πθ + πz
πθ + πz
Because market clearing in period 2 requires p = E [θ|K] , we conclude that the equilibrium price
satisfies
p(θ, y) = γ0 + γ1 K(θ, y),
where
γ0 ≡
πθ
πz
β0
µ−
πθ + πz
πθ + πz β1 + β2
and
These results are summarized in the following lemma.
9
γ1 ≡
(5)
πz
1
.
πθ + πz β1 + β2
(6)
Lemma 1 In any linear equilibrium, there are coefficients (β0 , β1 , β2 , γ0 , γ1 ) such that
k (x, y) = β0 + β1 x + β2 y
and
p (θ, y) = γ0 + γ1 K (θ, y) .
Moreover, γ1 > 0 if and only if β1 + β2 > 0.
The key result here is the relation between K and p. Provided that high investment is “good
news” for profitability (in the sense that a higher realization of K raises the traders’ expectation
of θ), financial prices increase with aggregate investment (γ1 > 0). This in turn induces strategic
complementarity in investment decisions. Indeed, when the entrepreneurs are choosing a higher
level of investment, they are sending a positive signal to the financial market, thus increasing the
price at t = 2. But then each entrepreneur’s willingness to invest at t = 1 is higher when he expects
a higher level of investment from other entrepreneurs, which means precisely that investment choices
are strategic complements. We formalize these intuitions in the next result, which follows directly
from replacing condition (5) into condition (2).
Lemma 2 In any linear equilibrium, the investment strategy satisfies
k (x, y) = E[(1 − α)κ (θ) + αK (θ, y) | x, y ],
where α ≡ λγ1 and κ (θ) ≡
(7)
(1−λ)θ+λγ0
.
1−λγ1
Condition (7) can be interpreted as the best-response condition in the coordination game that
emerges among the entrepreneurs for a given price function: it describes the optimal strategy for
each individual entrepreneur as a function of his expectation of aggregate investment (the relevant
summary of the strategy of other entrepreneurs), taking as given the impact of the latter on financial
prices. The coefficient α then measures the degree of strategic complementarity in investment
decisions: the higher α, the higher the slope of the best response of individual investment to
aggregate investment, that is, the higher the incentive of entrepreneurs to align their investment
choices. The function κ(θ), on the other hand, captures the impact of the fundamental on the
individual return of investment for given K, normalized by 1 − α.9
A similar best-response condition characterizes the class of linear-quadratic games examined in
Angeletos and Pavan (2007a), including the special case of Morris and Shin (2002). However, there
are two important differences. First, while in those games the degree of strategic complementarity α
is exogenously determined by the payoff structure, here it is endogenously determined as an integral
9
This normalization serves two purposes. First, it identifies κ(θ) with the complete-information equilibrium level of
investment in the game among the entrepreneurs, for given price function. Second, it ensures that the unconditional
mean of investment is given by Ek(x, y) = Eκ(θ).
10
part of the equilibrium. Second, while in those games the degree of complementarity is independent
of the information structure, here it actually originates in the dispersion of information. In fact, the
complementarity in our setup is solely due to the informational content of aggregate investment.
How much information aggregate investment conveys about θ determines the coefficient γ1 , which
captures the sensitivity of prices to aggregate investment. In turn, the coefficient γ1 pins down
the value of α, which captures the degree of complementarity in the entrepreneurs’ investment
decisions. In the absence of informational frictions (the benchmark case examined in the previous
section), aggregate investment provides no information to the traders, prices are thus independent
of K, and the complementarity in investment decisions is absent. When instead information is
dispersed, aggregate investment becomes a signal of θ, prices respond to aggregate investment and
a complementarity in investment decisions emerges. The more informative aggregate investment is
about θ, the stronger the complementarity.
Because the complementarity depends on the informational content of aggregate investment,
which in turn depends on the entrepreneurs’ strategies, to determine the equilibrium value of
α we need to solve a fixed-point problem. Before doing so, we first show how this endogenous
complementarity is instrumental in understanding the incentives entrepreneurs face in using their
available sources of information.
Lemma 3 In any linear equilibrium,
β2
πy 1
=
.
β1
πx 1 − α
(8)
Therefore, provided that β1 , β2 > 0, the sensitivity of the entrepreneurs’ equilibrium strategy to the
common signal relative to the idiosyncratic signal is higher the higher the equilibrium degree of
complementarity.
Let us provide some intuition for this result. Consider an entrepreneur’s best response to the
strategy that other entrepreneurs follow, holding fixed the price function. Suppose that the other
entrepreneurs’ strategy is k (x, y) = β0 + β1 x + β2 y, with β1 , β2 > 0.10 Aggregate investment is then
given by K (θ, y) = β0 + β1 θ + β2 y and an agent’s best predictor of aggregate investment is
E[K|x, y] = β0 + β1 E[θ|x, y] + β2 y.
10
This condition means that investment responds positively to both signals. Since an entrepreneur’s expectation of
θ increases with either signal, it is quite natural to expect this condition to be satisfied in equilibrium. Below we will
show that an equilibrium that satisfies this condition always exists and that this equilibrium is unique for λ small
enough. For λ high enough, however, it is possible to construct equilibria in which entrepreneurs find it optimal to
react negatively to a signal because they expect others to do the same.
11
The private signal x helps predict aggregate investment only through E[θ|x, y], while the common
signal y helps predict aggregate investment both through E[θ|x, y] and directly through its effect
on the term β2 y. Therefore, relative to how much the two signals help predict the fundamental,
the common signal y is a better predictor of aggregate investment than the private signal x. But
now recall that a higher α means a stronger incentive for an individual entrepreneur to align his
investment choice with that of other entrepreneurs. It follows that when α is higher entrepreneurs
find it optimal to rely more heavily on the common signal y relative to the private signal x, for it
is the former that best helps them align their choice with the choice of others.11
3.2
Equilibrium characterization
As noted earlier, completing the equilibrium characterization requires solving a fixed-point problem.
On the one hand, how entrepreneurs use their available information depends on α, the endogenous
complementarity induced by the response of prices to aggregate investment. On the other hand,
how sensitive asset prices are to aggregate investment, and hence how strong α is, depends on
how informative aggregate investment is about the fundamental, which in turn depends on how
entrepreneurs use their available information in the first place. This fixed-point problem captures
the essence of the two-way feedback between the real and the financial sector in our model. Its
solution is provided in the following lemma.
Lemma 4 There exist functions F : R × (0, 1) × R3+ → R and G : R × (0, 1) × R3+ → R5 such that
the following are true:
(i) In any linear equilibrium, β2 /β1 solves
β2
=F
β1
β2
; λ, πθ , πx , πy
β1
(9)
while (β0 , β1 , β2 , γ0 , γ1 ) = G (β2 /β1 ; λ, πθ , πx , πy );
(ii) Equation (9) has at least one solution at some β2 /β1 > πy /πx ;
(iii) For any (πθ , πx , πx ), there exists a cutoff λ̄ = λ̄ (πθ , πx , πy ) > 0 such that (9) admits a
unique solution if λ < λ̄;
(iv) There exists an open set S such that (9) admits multiple solutions if (λ, πθ , πx , πx ) ∈ S.
The fixed-point problem that leads to the equilibrium characterization is set up in terms of
the variable b = β2 /β1 , which represents the relative sensitivity of entrepreneurial investment to
the two signals. Given b, we can determine the sensitivity of the price to aggregate investment γ1 .
11
A similar property holds for the more general information structures considered in the Supplementary Material: a
stronger complementarity shifts the use of information towards the signals whose errors are relatively more correlated
across agents.
12
Given γ1 , we can then determine the complementarity α and then the sensitivity b of individual best
responses to the two signals. These steps describe the mapping F used in Lemma 4 and provide
the intuition for part (i) of the lemma: the fixed points of F identify all the linear equilibria of
our economy. Parts (ii)-(iv) then characterize the fixed points of F , establishing than a linear
equilibrium always exist and, although it is not always unique, it is unique for λ small enough.
The possibility of multiple equilibria for high values of λ is interesting for several reasons. First,
it illustrates the potential strength of the two-way feedback between real and financial activity. Second, this multiplicity originates solely from an informational externality rather than from the more
familiar payoff effects featured in coordination models of crises à la Diamond and Dybvig (1984)
and Obstfeld (1996). Finally, this multiplicity can induce additional non-fundamental volatility in
both real investment and financial prices.
However, the possibility of multiple equilibria is not central to our analysis. When there is a
unique equilibrium, the key positive and normative predictions documented in Corollaries 1 and 2
below necessarily hold. When there are multiple equilibria, these predictions continue to hold for
any equilibrium that satisfies the natural property that investment increases with both signals. For
the rest of the paper we thus leave aside the possibility of multiplicity and focus on the case where
the equilibrium is unique. The next proposition then summarizes some key equilibrium properties.
Proposition 1 There always exists a linear equilibrium in which the following properties are true:
(i) Individual investment increases with both signals (β1 , β2 > 0) and hence the equilibrium
price increases with aggregate investment (γ1 > 0);
(ii) The equilibrium degree of complementarity satisfies 0 < α < 1 and is increasing in λ;
(iii) The sensitivity of investment to the common signal relative to the private is higher than
πy /πx and is increasing in λ.
Moreover, λ small enough suffices for this equilibrium to be the unique linear equilibrium.
Part (i) guarantees that individual investment increases with both signals, which in turn ensures
that high aggregate investment is necessarily “good news” for profitability and hence that α is
positive. Part (ii) further establishes that α is higher the stronger the entrepreneurs’ concern
about financial prices. Combining this with Lemma 3 then gives part (iii), which spells out the
implications for the equilibrium use of information.
3.3
Impact of fundamental and expectational shocks
To further appreciate the positive implications of informational frictions—and the complementarity
thereof—it is useful to rewrite aggregate investment as
K = β0 + (β1 + β2 ) θ + β2 ε.
13
Aggregate investment thus depends on two types of shocks: fundamental shocks, captured by θ,
and expectational shocks, captured by ε. How entrepreneurs use available information affects how
investment respond to these shocks: the sensitivity to fundamentals is governed by the sum β1 + β2 ,
while the sensitivity to expectational shocks is governed by β2 .
When information is dispersed, prices react positively to aggregate investment, the equilibrium
degree of complementarity is positive, and hence the relative sensitivity to the common signal
satisfies β2 /β1 > πy /πx . In contrast, when there are no informational frictions, prices do not react to
aggregate investment, the equilibrium degree of complementarity is zero, and hence β2 /β1 = πy /πx .
The following is then an immediate implication.
Corollary 1 (Main positive prediction) The impact of expectational shocks relative to fundamental shocks is higher in the presence of informational frictions.
This result is the key positive prediction of the paper: informational frictions amplify nonfundamental volatility relative to fundamental volatility; that is, they reduce the R-square of a
regression of aggregate investment on expected profits. Importantly, because the equilibrium α
increases with λ, this amplification effect is stronger the more entrepreneurs care about asset
prices.12
Corollary 1 regards the relative impact of the two shocks. The next proposition reinforces this
finding by examining the absolute impact of the two shocks.
Proposition 2 There exists λ̂ > 0 such that, for all λ ∈ (0, λ̂], there is a unique linear equilibrium
and the following comparative statics hold:
(i) higher λ reduces β1 + β2 , thus dampening the impact of fundamental shocks;
(ii) higher λ increases β2 , thus amplifying the impact of expectational shocks.
The key intuition for these results is again the role of the complementarity for the equilibrium
use of information. To see this, suppose for a moment that γ0 = 0 and γ1 = 1, meaning that
p (θ, y) = K (θ, y) in all states. The entrepreneurs’ best response then reduces to
k (x, y) = E [ (1 − λ) θ + λK (θ, y) | x, y ] ,
12
(10)
This result extends to the more general information structures considered in the Supplementary Material. Because
all signals have common errors, there are multiple “expectational shocks.” We can then decompose the total variance
of investment in two components: the one that is explained by θ (which defines “fundamental volatility”), and the
one that is explained by the combination of all common errors (which defines “non-fundamental volatility”). We then
show that the ratio of the latter to the former is higher with dispersed information (in which case α > 0) than in the
frictionless benchmark (in which case α = 0).
14
so that the degree of complementarity now coincides with λ. One can then easily show that the
unique solution to (10) is k (x, y) = β0 + β1 x + β2 y, with
β0 =
πθ
µ,
πθ + πx (1 − λ) + πy
β1 =
πx (1 − λ)
,
πθ + πx (1 − λ) + πy
and β2 =
πy
.
πθ + πx (1 − λ) + πy
It is then immediate that a higher λ increases the sensitivity to the prior (captured by β0 ) and
the sensitivity to the common signal (captured by β2 ), while it decreases the sensitivity to the
private signal (captured by β1 ). We have already given the intuition for the result that a stronger
complementarity amplifies the reliance on the common signal and dampens the reliance on the
private signal. That it also increases the reliance on the prior is for exactly the same reason as for
the common signal: the prior is a relative good predictor of others’ investment choices.
However, note that the average return on investment coincides with the mean of θ, irrespective
of λ. This is because the average price must equal the mean of θ, for otherwise the traders would
make on average non-zero profits, which would be a contradiction. But then the average investment
must also be equal to the average of θ, that is, β0 + (β1 + β2 ) µ must equal µ. It is then immediate
that, because a higher λ increases β0 , it also reduces the sum β1 + β2 . In simple words, investment
is less sensitive to changes in fundamentals simply because the complementarity strengthens the
anchoring effect of the prior.
These intuitions would be exact if γ0 = 0 and γ1 = 1, or more generally if these coefficients
were exogenous to λ. In our model, the price is an increasing function of aggregate investment, but
the coefficients γ0 and γ1 depend on λ. This explains why these intuitions are incomplete and why
the absolute effects documented in Proposition 2 hold only for a subset of the parameter space.
However, the prediction regarding the amplification of the relative impact of non-fundamental
shocks (Corollary 1) holds true more generally.
4
Constrained efficiency
The analysis so far has focused on the positive properties of the equilibrium. We now study
its normative properties by examining whether there is an allocation that, given the underlying
information structure, leads to higher welfare.
The question of interest here is whether society can do better, relative to equilibrium, by having
the agents use their available information in a different way—not whether society can do better
by giving the agents more information. We thus adopt the same constrained efficiency concept
as in Angeletos and Pavan (2007a,b): we consider the allocation that maximizes ex-ante welfare
subject to the sole constraint that the choice of each agent must depend only on the information
available to that agent. In other words, we let the planner dictate how agents use their available
15
information, but we do not let the planner transfer information from one agent to another. In so
doing, we momentarily disregard incentive constraints; later on we will identify tax systems that
implement the efficient allocation as an equilibrium.
Note that the payments in the financial market represent pure transfers between the entrepreneurs and the traders and therefore do not affect ex-ante utility.13 We can thus focus on
the investment strategy and define the efficient allocation as follows.
Definition 3 The efficient allocation is a strategy k(x, y) that maximizes ex-ante utility
Eu =
with K (θ, y) =
Z Z
R
1
2
h
(1 − λ) θk (x, y) −
2
1
2 k (x, y)
i
dΦ(x|θ) + [θλK (θ, y)] dΨ(θ, y)
1
2
(11)
k (x, y) dΦ(x|θ).
Condition (11) gives ex-ante utility for an arbitrary strategy. The first term in square brackets
is the payoff of an entrepreneur with information (x, y); the second term in square brackets is
the payoff of a trader when aggregate investment is K(θ, y); finally Ψ denotes the cumulative
distribution function of the joint distribution of (θ, y). Note that the transfer of capital from the
entrepreneurs that are hit by the liquidity shock to the traders does not affect the return to
investment. It follows that (11) can be rewritten compactly as14
Eu = 12 E[V (k (x, y) , θ) ] = 12 E[E[V (k (x, y) , θ) |x, y]],
where V (k, θ) ≡ θk − 12 k2 . From the society’s viewpoint, λ is irrelevant and it is as if the entrepreneurs’ payoffs are V (k, θ). It is then obvious that a strategy k(x, y) is efficient if and only if,
for almost all x and y, k(x, y) maximizes E[V (k, θ)|x, y]. The following result is then immediate.
Proposition 3 The efficient investment strategy is given by
k (x, y) = E [θ|x, y] = δ0 µ + δ1 x + δ1 y,
where
δ0 ≡
πθ
,
πθ + πx + πy
δ1 ≡
πx
,
πθ + πx + πy
δ2 ≡
πy
.
πθ + πx + πy
13
By ex-ante utility, we mean before the realization of any random variable, including those that determine whether
an agent will be an entrepreneur or a trader. However, note that, because utility is transferable, any strategy k(x, y)
that maximizes ex-ante utility also maximizes any weighted average of the expected utility of an entrepreneur and of
a trader. By the same token, any strategy k(x, y) that improves upon the equilibrium in terms of ex-ante utility can
yield a Pareto improvement. It suffices, for example,
to let the entrepreneurs and the traders continue to trade at a
R
price p(θ, y) = E(θ|K(θ, y)), where K(θ, y) = k (x, y) dΦ(x|θ).
14
It suffices to substitute the expression for K (θ, y) in (11).
16
Note that the efficient strategy would have coincided with the equilibrium strategy if it were
not for informational frictions. It follows that our earlier positive results admit a normative interpretation.15
Corollary 2 (Main normative prediction) In the presence of informational frictions, the impact of expectational shocks relative to fundamental shocks is inefficiently high.
5
Policy implications
Having identified a potential source of inefficiency, we now analyze the effect of different policies.
First, we consider interventions “during the fact,” by which we mean interventions in the financial
market at t = 2, when uncertainty about θ has not been resolved yet. Next, we consider policies
“after the fact,” by which we mean policies contingent on information that becomes public at t = 3,
after uncertainty about θ has been resolved. In both cases, we impose that the government has no
informational advantage vis-a-vis the private sector, which is our preferred benchmark for policy
analysis. At the end of this section, however, we also consider situations where the government can
directly affect the information available to the agents.
5.1
Interventions “during the fact”: price stabilization
We start by considering policies aimed at reducing asset-price volatility. In particular, suppose the
government imposes a proportional tax τ on financial trades (purchases of capital) at date 2. This
tax can depend on the price, which is public information. For simplicity, it takes the following
linear form:
τ (p) = τ0 + τ1 p,
(12)
where (τ0 , τ1 ) are scalars. Tax revenues are rebated as a lump sum.
The equilibrium price in the financial market is now given by p = E [θ|K] − (τ0 + τ1 p) ; equivalently,
p=
1
1
(E [θ|K] − τ0 ) =
(γ0 + γ1 K − τ0 ) ,
1 + τ1
1 + τ1
(13)
where γ0 and γ1 are given, as before, by (6). When the tax is pro-cyclical (i.e., τ1 > 0), its
effect is to dampen the response of asset prices to the traders’ expectation of θ, and thereby their
response to the news contained in aggregate investment. In equilibrium, this tends to reduce the
degree of complementarity in investment decisions. To see this more clearly, note that the degree
15
Corollary 2 presumes that the equilibrium is unique. When there are multiple equilibria, the result holds for any
equilibrium in which β1 , β2 > 0. Since the efficient allocation satisfies β1 , β2 > 0, this also ensures that no equilibrium
is efficient.
17
Welfare
−0.012
−0.014
−0.016
−0.018
−0.1
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
0.21
1
0.205
β
0.2
0.195
0.19
−0.1
0.5
β
2
0.45
0.4
0.35
0.3
−0.1
τ
1
Figure 1: The impact of price-stabilization policies.
of complementarity is now given by
α=
λγ1
.
1 + τ1
If γ1 were exogenous, it would be immediate that α decreases with τ1 . But if α falls, we know from
Lemma 3 that β2 /β1 , the relative weight on common sources of information, must also fall. This
in turn means that aggregate investment becomes more informative about θ, so that γ1 increases,
counteracting the direct effect of τ1 on α. However, one can show that, at least as long as the
equilibrium is unique, the direct effect dominates, guaranteeing that the equilibrium degree of
complementarity decreases with τ1 .16
We conclude that a higher τ1 , by reducing the degree of complementarity, necessarily reduces
the relative impact of expectational shocks. However, by reducing the overall sensitivity of prices
to all sources of variation in investment, a higher τ1 also reduces the impact of fundamental shocks.
As argued in the previous section, in the absence of policy intervention, investment is excessively
sensitive to expectational shocks and insufficiently sensitive to fundamental shocks. It follows that
the welfare consequences of the tax are ambiguous: while reducing the impact of expectational
shocks improves efficiency, reducing the impact of fundamental shocks has the opposite effect.
These intuitions are illustrated in Figure 1 where for each value of τ1 , the value of τ0 is chosen
16
This follows from an argument similar to the one that establishes that α is monotonic in λ.
18
optimally to maximize welfare. The top panel depicts the difference in welfare under the stabilization policy considered here and under the constrained efficient allocation; the bottom panels depict
the sensitivity to expectational shocks β2 and to fundamental shocks (β1 + β2 ).17 The figure is
drawn for a baseline set of parameters: πθ = πx = πy = 1 and λ = 0.5. However, its qualitative
features are robust across a wide set of parametrizations. In particular, we have randomly drawn
10,000 parameter vectors (λ, πθ , πx , πy ) from (0, 1) × R3+ . For each such vector, we have found
that the optimal τ1 is positive and it induces a lower β2 and a lower β1 + β2 as compared to the
equilibrium without policy, reflecting the trade off discussed above.
While these numerical results, which span the entire parameter space, reveal that the optimal
policy always involves a strictly positive degree of price stabilization (i.e., τ1 > 0), we have not been
able to establish this result formally. However, it is easy to show that full price stabilization (i.e.,
τ1 → ∞) is never optimal. In this limit, prices cease to react to aggregate investment, the strategic
complementarity disappears, and equilibrium investment reduces to k(x, y) = (1 − λ) E[θ|x, y]. By
implication, the relative sensitivity of investment to expectational shocks β2 /(β1 + β2 ) is at its
efficient level, but its overall sensitivity to the fundamental is λ times lower than at the efficient
level. At this point, a marginal increase in the relative sensitivity implies only a second-order
welfare loss, while a marginal increase in the overall sensitivity implies a first-order welfare gain.
It follows that it is never optimal to fully stabilize the price.
Proposition 4 A tax that stabilizes prices can increase welfare; however, a tax that completely
eliminates price volatility is never optimal.
5.2
Interventions “after the fact”: corrective taxation
Suppose now that the government imposes a proportional tax τ on asset holdings in period 3. The
tax is now paid by the entrepreneurs not hit by the liquidity shock and by the traders who acquired
capital in period 2. The advantage of introducing a tax in period 3 is that the tax rate τ can now
be made contingent on all information which is publicly available in that period, including K and
θ. We focus on linear tax schemes of the form
τ (θ, K) = τ0 + τ1 θ + τ2 K,
(14)
where (τ0 , τ1 , τ2 ) are scalars; tax revenues are again rebated in a lump-sum fashion. The following
result shows that these simple tax schemes can implement the constrained efficient allocation.
17
Note that τ0 affects the unconditional average of k (x, y), but has no effect on the sensitivity of investment to the
signals x and y, i.e., on β1 and β2 . We henceforth concentrate on τ1 .
19
Proposition 5 There exists a unique linear tax scheme that implements the efficient allocation as
an equilibrium. The optimal tax satisfies τ0 < 0, τ1 < 0 and τ2 > 0.
The intuition behind this result is that the government can control the degree of strategic
complementarity perceived by the agents by appropriately designing the contingency of the marginal
tax rate τ on aggregate investment: the higher the elasticity τ2 of the marginal tax rate with respect
to K, the lower the degree of complementarity in investment choices α and the lower the sensitivity
of equilibrium investment to common noise relative to idiosyncratic noise. This effect is analogous
to that of the stabilization policies discussed above. However, the government now has an extra
instrument available: the elasticity τ1 of the tax to the realized fundamental. Through τ2 the
government can thus induce the optimal relative sensitivity to expectational shocks β2 /(β1 + β2 )
while, at the same time, adjust τ1 to obtain the optimal absolute sensitivities to each shock.18
Although this result does not require any informational advantage on the government’s side, it
assumes that the government observes perfectly the fundamental θ and the agents’ capital holdings
at the time taxes are collected. However, the result easily extends to situations where these quantities are observed with measurement error. In particular, suppose that in stage 3 the government
only observes θ̃ = θ + ǫ and s̃i = si + ηi for each i, where si is the capital holding of agent i,
while ǫ and ηi are measurement errors, possibly correlated with one another, but independent of
θ and of the agents’ information in period 2. Then let K̃ be the cross-sectional mean of s̃i (i.e.,
the government’s measure of aggregate investment) and consider a proportional tax on s˜i of the
following form: τ (θ̃, K̃) = τ0 + τ1 θ̃ + τ2 K̃. It is then easy to check that there continues to exist a
unique set of coefficients (τ0 , τ1 , τ2 ) that implement the efficient allocation as an equilibrium and
that these coefficients continue to satisfy τ1 < 0 < τ2 .
To recap, the key insight here is that the government can use the contingency of the tax rate on
public signals of θ and K that will be revealed at stage 3 to achieve efficiency in the decentralized use
of information during stage 1. Although this information becomes available only after all investment
decisions are sunk, by promising a specific policy response to this information the government is
able to manipulate how entrepreneurs use their available sources of information when making their
investment decisions, even if it cannot directly monitor these sources of information.19
18
The optimal τ0 is then chosen to induce the optimal level of unconditional average investment. Similar tax
schemes implement the efficient investment strategy in all the extensions considered in Section 6.
19
These intuitions, and the implementation result in Proposition 5, build on Angeletos and Pavan (2007b). This
paper considers optimal policy within a rich, but abstract, class of economies with dispersed information on correlated
values. See also Lorenzoni (2007) for monetary policy in a business-cycle model with dispersed information on
underlying productivity.
20
5.3
Optimal release of information
We now turn to policies that affect the information available to the agents. This seems relevant
given the role of the government in collecting (and releasing) macroeconomic data.
To capture this role, suppose that in stage 2 traders can only observe average investment with
noise, that is, they observe
K̃ = K + η,
where η is aggregate measurement error, which is a random variable, independent of all other shocks,
with mean zero and variance 1/πη .20 Suppose now that the government can affect the precision πη of
the macroeconomic data available to financial traders. By changing πη the government determines
the weight that traders assign to K̃ when estimating future profitability. This is another channel
by which the government is able to affect the degree of strategic complementarity in investment
decisions.
Indeed, the choice of πη is formally equivalent to the choice of τ1 in the setup with a tax on
financial transactions (Section 5.1). For each value πη of the precision of the signal about aggregate
activity, there is a value τ1 of the tax elasticity that induces the same equilibrium strategy, and vice
versa. To see this, note that in any linear equilibrium, the observation of K̃ = β0 +(β1 +β2 )θ+β2 ε+η
is informationally equivalent to the observation of a signal
z≡
K̃ − β0
β2
1
=θ+
ε+
η,
β1 + β2
β1 + β2
β1 + β2
with precision
πz =
β2
β1 + β2
2
πy−1
+
1
β1 + β2
2
πη−1
!−1
.
The equilibrium price is then given by p(θ, y, η) = γ0 + γ1 [K(θ, y) + η], with γ0 and γ1 given by (6),
and hence the degree of strategic complementarity remains equal to α ≡ λγ1 , as in the baseline
model. By changing the value of πη , the government can then directly manipulate γ1 and thus the
degree of strategic complementarity perceived by the entrepreneurs.21
We conclude that the choice of πη is subject to the same trade-offs emphasized for the choice
of τ1 : decreasing πη reduces the relative response of investment to expectational shocks, but it also
reduces its response to fundamental shocks. The results of Section 5.1 then imply that an intermediate degree of release of macroeconomic data may be optimal even when the cost of collecting
such data is zero.22
20
The equilibrium characterization for this case is a straightforward extension of the baseline case.
See the Supplementary Material for the proof of this claim.
22
Note, however, that this holds only as long as the equilibrium is inefficient. If, instead, the policies considered
in Section 5.2 are in place, thus guaranteeing that the equilibrium is efficient, then a higher πη is always welfare
21
21
Finally, one could consider policies which affect directly the agents’ information regarding the
fundamental θ. In particular, the government can collect some information about θ in period 1 and
decide whether to disclose this information to the entrepreneurs, or to both the entrepreneurs and
the traders. In the first case, the policy corresponds to an increase in the precision of the signal y in
the baseline model. Although entrepreneurs have a more precise estimate of the fundamental, this
information is not shared with the traders. Therefore, this policy could exacerbate the asymmetry
of information and could magnify the feed-back effects between investment and asset prices, with
possible negative consequences on social welfare.
In the second case, instead, the policy corresponds to an increase in the precision of the common
prior in the baseline model. This policy is socially beneficial for two reasons: first, it improves the
quality of the information available to the entrepreneurs and hence it permits them to better align
their decisions to the fundamental. Second, it reduces the reliance of financial markets on the
endogenous signal K in their estimate of the fundamental. This second effect tends to reduce
the degree of strategic complementarity in investment decisions, and hence also the discrepancy
between equilibrium and efficient allocations. Both effects then contribute to higher welfare.
6
Extensions
Our analysis has identified a mechanism through which the dispersion of information induces complementarity in real investment choices, amplification of non-fundamental disturbances, and inefficiency of market outcomes, all at once. In the baseline model, we have made a number of
assumptions to illustrate this mechanism in the simplest possible way. In particular, we have assumed that the traders’ demand for installed capital is perfectly elastic, that entrepreneurs who
are not hit by the liquidity shock do not trade in the financial market, and that the traders’ valuation of the asset coincides with that of the entrepreneurs. In this section, we relax each of these
assumptions.
We first extend the model to allow for the traders’ demand for capital to be downward sloping.
This extension is interesting because it introduces a potential source of strategic substitutability
in the entrepreneurs’ investment decisions: when aggregate investment is higher, the supply of
installed capital in the financial market is also higher, putting a downward pressure on asset prices
and lowers the ex-ante incentive to invest.
In a second extension, we allow entrepreneurs not hit by the liquidity shock to participate in
the financial market. This extension is interesting for two reasons: first, it allows for some of
the entrepreneurs’ information to be aggregated in the financial market; second, it introduces a
non-trivial allocative role for prices.
improving.
22
Although some interesting differences arise, the key positive and normative predictions of the
paper (Corollaries 1 and 2) remain valid in both extensions: as long as the dispersion of information
does not completely vanish in the financial market, the signaling effect of aggregate investment
continues to be the source of amplification and inefficiency in the response of the equilibrium to
common sources of noise.
Finally, we consider a variant that introduces shocks to the financial-market valuation of the
installed capital. This variant brings the paper closer to the recent literature on “mispricing”
and “bubbly” asset prices. It also helps clarify that the details of the information structure we
assumed in the baseline model are not essential: any source of common noise in the information that
aggregate investment conveys about the underlying fundamentals opens the door to amplification
and inefficiency.
6.1
The supply-side effect of capital: a source of strategic substitutability
We modify the benchmark model as follows. The net payoff of trader i, who buys qi units of capital
at the price p, is now given by
ui = (θ − p) qi −
1 2
q ,
2φ i
(15)
where φ is a positive scalar. The difference with the benchmark model is the presence of the last
term in (15), which represents a transaction cost associated to the purchase of qi units of capital. A
convex transaction cost ensures a finite price elasticity for the traders’ demand, which is now given
by q(p, K) = φ (E [θ|K] − p). The parameter φ captures the price elasticity of this demand function
and our benchmark model corresponds to the special case where the demand is infinitely elastic,
i.e. φ → ∞.23 As in the benchmark model, in any linear equilibrium the traders’ expectation of θ
is given by E[θ|K] = γ0 + γ1 K, for some coefficients γ0 and γ1 . However, unlike in the benchmark
model, the equilibrium price does not coincide with E[θ|K]. Market clearing now requires that
q(p, K) = λK, so the equilibrium price is
p = E[θ|K] − λφ K = γ0 + γ1 − φλ K.
(16)
It follows that aggregate investment has two opposing effects on the price of installed capital, p.
On the one hand, it raises the traders’ expectation of θ, thereby pushing the price up. On the
other hand, it raises the supply of capital, thereby pulling the price down. The strength of these
two effects determines whether investment choices are strategic complements or substitutes.
23
A more familiar way of introducing a finitely elastic demand is to assume risk aversion. The alternative we use
here captures the same key positive and normative properties—namely, demands are finitely elastic and individual
payoffs are concave in own portfolio positions—but has the advantage of keeping the analysis tractable by making
the elasticity of demands invariant to the level of uncertainty.
23
Proposition 6 (i) In any linear equilibrium, the investment strategy satisfies
k (x, y) = E [ (1 − α) κ (θ) + αK (θ, y) | x, y ] ,
with α ≡ λγ1 − λ2 /φ and κ (θ) ≡
(17)
(1−λ)θ+λγ0
1−λγ1 +λ2 /φ .
(ii) λ small enough suffices for the equilibrium to be unique, for investment to increase with θ,
and for γ1 to be positive.
The degree of complementarity α is now the sum of two terms. The first term λγ1 captures the,
by now familiar, informational effect of investment on asset prices documented in the benchmark
model. The second term, −λ2 /φ, captures the simple supply-side effect that emerges once the
demand for the asset is finitely elastic. If either the information contained in aggregate investment
is sufficiently poor (low γ1 ) or the price elasticity of demand is sufficiently low (low φ), investment
choices become strategic substitutes (α < 0). However, the question of interest here is not whether
investment choices are strategic complements or substitutes, but how the positive and normative
properties of the equilibrium are affected by the dispersion of information. In this respect, the
implications that emerge in this extension are essentially the same as in the benchmark model.
First, consider the positive properties of the equilibrium. Lemma 3 immediately extends to the
modified model: equilibrium investment satisfies k(x, y) = β0 + β1 x + β2 y with
β2
πy 1
=
.
β1
πx 1 − α
(18)
Provided that investment increases with both signals, then aggregate investment is necessarily
good news for θ (i.e., γ1 > 0), in which case Proposition 6 implies that α > −λ2 /φ. In contrast,
when there are no informational frictions, the equilibrium price is simply p = θ − (λ2 /φ)K, so
that α = −λ2 /φ. It follows that the dispersion of information increases the value of α and hence
amplifies the impact of common expectational shocks relative to that of fundamental shocks, even
if α happens to be negative. We conclude that Corollary 1, which summarizes the key positive
predictions of the model, continues to hold.
Next, consider the normative properties. Because of the convexity of the transaction costs, it
is necessary for efficiency that all traders take the same position in the financial market: qi = λK
for all i ∈ (1/2, 1]. Ex-ante utility then takes the form
Eu =
=
Z Z
1
2
(1 − λ) θk(x, y) −
2
1
2 k(x, y)
2
1
k
1
E − + θk −
(λK)2
2
2
2φ
dΦ(x|θ) +
24
1
2
h
θλK(θ, y) −
2
1
2φ [λK(θ, y)]
i
dΨ(θ,
(19)
y)
and the efficient investment strategy is the function k(x, y) that maximizes (19).
Proposition 7 The efficient investment strategy is the unique linear solution to
k (x, y) = E [(1 − α∗ ) κ∗ (θ) + α∗ K (θ, y) | x, y ] ,
(20)
R
where α∗ ≡ −λ2 /φ < 0, κ∗ (θ) ≡ θ/ 1 + λ2 /φ , and K(θ, y) = k(x, y)dΦ(x|θ).
To understand this result, note that the social return to investment is given by (1 − λ) θ +
λ (θ − λK/φ) = θ − λ2 K/φ. The new term, relative to the benchmark model, is −λ2 K/φ and
it reflects the cost associated with transferring λK units of the asset from the entrepreneurs to
the traders. If information were complete, efficiency would require that each agent equates his
marginal cost of investing to the social return to investment, which would give k = θ − λ2 K/φ.24
The analogue under incomplete information is that each agent equates the marginal cost to the
expected social return:
k (x, y) = E
Rearranging this condition gives (20).
θ − λ2 /φ K (θ, y)
x, y .
(21)
The key finding here is that the introduction of downward sloping demands has a symmetric
effect on the private and social returns to investment. This is simply because the negative pecuniary
externality caused by the higher supply of capital perfectly reflects the social cost associated with
having traders absorb this additional capital. As a result, it is only the informational effect that
generates a discrepancy between the equilibrium and the efficient allocation.
As in the benchmark model, this discrepancy manifests itself in the response of equilibrium to
expectational and fundamental shocks. Indeed, while equilibrium investment satisfies (18), efficient
investment satisfies k(x, y) = β0∗ + β1∗ x + β2∗ y with
β2∗
πy 1
=
.
∗
β1
πx 1 − α∗
Because in any equilibrium in which β1 , β2 > 0 the complementarity satisfies α∗ < α < 1, the
relative sensitivity of the equilibrium strategy to common noise is inefficiently high. We conclude
that Corollary 2, which summarizes the key normative predictions of the model, continues to hold.25
24
Note that under full information the optimal level of investment would be equal to κ∗ (θ).
As common in competitive environments, there are other forms of pecuniary externalities that could induce
strategic substitutability in the entrepreneurs’ investment decisions, even with a perfectly elastic demand for capital.
For example, suppose that, in order to complete their investment, entrepreneurs need to purchase certain inputs
whose aggregate supply is imperfectly elastic (e.g., labor or land). Higher aggregate investment then implies higher
aggregate demand for these inputs, and hence higher input prices and lower entrepreneurial returns, once again
inducing strategic substitutability in the entrepreneurs’ investment choices. However, such pecuniary externalities do
not, on their own, cause discrepancies between private and social returns. Indeed, it is easy to construct variants of
25
25
6.2
Information aggregation through prices
The analysis so far has imposed that the entrepreneurs who are not hit by the liquidity shock
can not access the financial market. Apart from being unrealistic, this assumption rules out the
possibility that the price in the financial market aggregates, at least partly, the information that
is dispersed among the entrepreneurs. To address this possibility, in this section we extend the
analysis by allowing entrepreneurs not hit by the liquidity shock to participate in the financial
market.
To guarantee downward sloping demands, we assume that traders and entrepreneurs alike incur
a transaction cost for trading in the financial market of the same type as in the previous section.26
Thus, the payoff of an entrepreneur i who is not hit by a liquidity shock, has invested ki units in
the first period, and trades qi units in the second period, is given by
1
1 2
q + θ (ki + qi ) ,
ui = − ki2 − pqi −
2
2φ i
while the payoff of a trader i is given by (15), as in the previous section.
Because the observation of K in the second period perfectly reveals θ to every entrepreneur,27
their demand for the asset in the second period reduces to qE = φ (θ − p). The demand of the
traders, on the other hand, is given by qT = φ (E [θ|K, p] − p). Note that traders now form their
expectation of θ based on K and on the information revealed by the equilibrium price p.28 Because
the aggregate demand for the asset is
1
2
(1 − λ) qE + 12 qT and the aggregate supply is 12 λK, market
clearing implies
p=
1
2−λ E [θ|K, p]
+
1−λ
2−λ θ
−
1
φ(2−λ) λK.
It follows that the joint observation of K and p perfectly reveals θ to the traders as well. The
1
asymmetry of information thus vanishes and the equilibrium price satisfies p = θ − φ(2−λ)
λK. As in
the previous section, this is just the social return to investment, adjusted for the fact that the total
capital of the entrepreneurs hit by the liquidity shock (λK/2) is now equally distributed among
the traders and the entrepreneurs not hit by the liquidity shock. Because the equilibrium price
coincides with the social return to investment it follows that the equilibrium is efficient.
This result is no different from what we established for the frictionless benchmark at the end
the model that capture such sources of strategic substitutability while retaining the property that the informational
effect of aggregate investment is the sole source of amplification and inefficiency, as in the example analyzed here.
26
We assume that the entrepreneurs hit by the liquidity shock do not pay the transaction cost for the units of the
asset that they have to sell in the second period; this simplification has no impact on the results.
27
This presumes that entrepreneurs use their private information when deciding how much to invest (i.e. β1 6= 0),
which is indeed true in equilibrium.
28
In the benchmark model, as well as in the extension examined in the previous section, we did not condition on
the information revealed by the equilibrium price simply because all agents trading voluntary in the financial market
had symmetric information.
26
of Section 2: if the dispersion of information vanishes at the time of trade in the financial market,
equilibrium investment is driven merely by first-order expectations of θ and is efficient. However,
this result hinges on the equilibrium price perfectly revealing θ. To make this clear, in the subsequent analysis we introduce an additional source of noise, which prevents prices from being perfectly
revealing.
Assume that the cost of trading for the entrepreneurs is subject to a shock ω, that is revealed
to them at the time they trade but which is not observed by the traders. In particular, the payoff
of an entrepreneur not hit by the liquidity shock is now given by
1
1 2
ui = − ki2 − pqi − ωqi −
q + θ (ki + qi ) ,
2
2φ i
where ω is assumed to be independent of all other random variables, with E[ω] = 0 and V ar[ω] =
σω2 ≡ πω−1 .
In what follows, we look at linear rational expectations equilibria; we continue to denote the
investment strategy by k (x, y) and we denote by p (θ, y, ω) the equilibrium price. Because the
observation of aggregate investment in the second period continues to reveal θ to the entrepreneurs
(but not to the traders), asset demands can be written as qE = φ (θ − ω − p) for the entrepreneurs
and qT = φ (E [θ|K, p] − p) for the traders. Market clearing then implies that the equilibrium price
is
p=
1
2−λ E [θ|K, p]
+
1−λ
2−λ
(θ − ω) −
1
φ(2−λ) λK.
(22)
Once again, the price is a weighted average of the traders’ and of the entrepreneurs’ valuation of the
asset, net of trading costs. However, because the shock ω is not known to the traders, the price no
longer perfectly reveals θ, ensuring that the informational effect of K on the traders’ expectation
of the fundamental reemerges. This effect is captured by the first term in the right-hand-side of
(22). At the same time, the supply-side effect of K is also present and is captured by the last term
in (22).
While the supply-side effect induces strategic substitutability, the informational effect induces
complementarity. In any rational expectations equilibrium in which the price is linear in (θ, y, ω),
E [θ|K, p] is the projection of θ on (K, p) and, by (22), the price p can be expressed as a linear combination of (K, θ, ω). It follows that, for any linear equilibrium, there exist coefficients (γ0 , γ1 , γ2 , γ3 )
such that29
E [θ|K, p] = γ0 + γ1 K + γ2 θ + γ3 ω.
(23)
29
Note that γ1 , which captures the effect of K on the traders’ expectation of θ, now combines the information that
is directly revealed to the traders by the observation of K with the information that is revealed to them through the
observation of the equilibrium price.
27
Using (23), (22), and the fact that the private return to investment is the expectation of (1 − λ) θ +
λp, we reach the following characterization result.
Proposition 8 (i) In any linear equilibrium, the investment strategy satisfies
k (x, y) = E[(1 − α) κ(θ) + αK (θ, y) | x, y],
where α =
λ
2−λ γ1
−
λ2
φ(2−λ)
and κ(θ) =
λγ0 +[2(1−λ)+λγ2 ]θ
2−λ−λγ1 +λ2 /φ .
(ii) λ small enough suffices for the equilibrium to be unique, for investment to increase with θ,
and for γ1 to be positive.
As in the previous section, α combines an informational effect (captured by
λ
2−λ γ1 )
with a
λ2
supply-side effect (captured by − φ(2−λ) ). The supply-side effect always contributes to strategic
substitutability, while the informational effect contributes to strategic complementarity if and only
if high investment is good news for θ (i.e. γ1 > 0). Once again, the overall effect is ambiguous, but
the role of informational frictions remains the same as before: Corollary 1 continues to hold.
We now turn to the characterization of the efficient allocation for this economy. The efficiency
concept we use is the same as in the preceding sections; however, now we need to allow the planner
to mimic the information aggregation that the market achieves through prices. We thus proceed
as follows.
First, we define an allocation as a collection of strategies k(x, y), qE (x, y, K, p, ω) and qT (K, p),
along with a shadow-price function p (θ, y, ω) with the following interpretation: in the first period,
an entrepreneur with signals (x, y) invests k (x, y); in the second period, all agents observe the
realizations of aggregate investment K = K (θ, y) and the shadow price p = p (θ, y, ω) ; the amount
of capital held by an entrepreneur not hit by a liquidity shock (in addition to the one chosen at
t = 1) is then given by qE (x, y, K, p, ω), while the amount of capital held by a trader is given by
qT (K, p).
Next, we say that the allocation is feasible if and only if, for all (θ, y, ω) ,
λK (θ, y) = (1 − λ)
Z
qE (x, y, K(θ, y), ω, p(θ, y, ω))dΦ (x|θ) + qT (K(θ, y), p(θ, y, ω)).
(24)
As with equilibrium, this constraint plays two roles: first, it guarantees that the second-period
resource constraint is not violated; second, it defines the technology that is used to generate the
endogenous public signal (equivalently, the extent to which information can be aggregated through
the shadow price).
Finally, for any given k(x, y), qE (x, y, K, p, ω) and qT (K, p), ex ante utility can be computed
28
as Eu = W (k, qE , qT ), where
W (k, qE , qT ) ≡
1
2
Z Z
1
{− k(x, y)2 + λp(θ, y, ω)k(x, y) + (1 − λ) θk(x, y) +
2
+ (1 − λ) R θ − ω, q E (x, y, K(θ, y), ω, p(θ, y, ω) }dΦ (x|θ) dΨ (θ, y, ω)
Z
1
+
R θ, q T (K(θ, y), p(θ, y, ω)) }dΨ (θ, y, ω) ,
2
where R (v, q) ≡ vq − q 2 / (2φ) and where K(θ, y) =
allocation as follows.
R
k(x, y)dΦ (x|θ) . We then define an efficient
Definition 4 An efficient allocation is a collection of strategies k(x, y), qE (x, y, K, p, ω) and qT (K, p),
along with a shadow price function p(θ, y, ω), that jointly maximize ex-ante utility, Eu = W (k, qE , qT ),
subject to the feasibility constraint (24).
Because utility is transferable, the shadow price does not affect payoffs directly; its sole function
is to provide an endogenous public signal upon which the allocation of the asset in the period 2
can be conditioned. The next lemma then characterizes the efficient allocation of the asset.
Lemma 5 The efficient allocation in the second period satisfies
∗
qE
=
λK
φω
−
2−λ 2−λ
and
qT∗ =
λK
(1 − λ) φω
+
.
2−λ
2−λ
(25)
To understand this result, suppose for a moment that information were complete in the second
period. For any given K, efficiency in the second period would require that all entrepreneurs hold
the same qE and that (qT , qE ) maximize
1 2
1 2
θqT −
q
+ (1 − λ) θqE − ωqE −
q
2φ T
2φ E
subject to (1 − λ) qE + qT = λK. Clearly, the solution to this problem is (25). In our environment,
information is incomplete but the same allocation can be induced through the following shadowprice and demand functions: p(θ, y, ω) = − φ(1−λ)ω
2−λ , qE (x, y, K, p, ω) =
λK
2−λ
−
λK
2−λ
−
φω
2−λ ,
and qT (K, p) =
p.30
We now characterize the efficient investment decisions. Using Lemma 5, ex ante utility reduces
to
1 2
1
(1 − λ)
1
2
Eu = E − k + θk −
(λK) +
φσ 2 .
2
2
2φ(2 − λ)
2 (2 − λ) ω
(26)
30
Note that the proposed shadow price is also the unique market-clearing price given the proposed demand functions.
The efficient trades can thus be implemented by inducing these demand functions through an appropriately designed
tax system and then letting the agents trade in the market.
29
Except for two minor differences—the smaller weight on (λK)2 , which adjusts the cost associated
with absorbing the fixed supply λK in the second period for the fact that now this quantity is
split across a larger pool of agents, and the last term in (26), which captures how the volatility
of ω affects the allocation of capital across entrepreneurs and traders in the second period—exante utility has the same structure as in (19) in the previous section. The following result is then
immediate.
Proposition 9 The efficient investment strategy is the unique linear solution to
k (x, y) = E [(1 − α∗ ) κ∗ (θ) + α∗ K (θ, y) | x, y] ,
2
λ
where α∗ ≡ − φ(2−λ)
, κ∗ (θ) ≡
1
1−α∗ θ,
and K(θ, y) =
R
(27)
k(x, y)dΦ(x|θ).
Comparing the efficient strategy with the equilibrium one, we have that, once again, as long as
investment increases with both signals, so that high investment is good news for profitability, then
α remains higher than α∗ , in which case the key normative prediction of the paper, as summarized
by Corollary 2, continues to hold.
6.3
Financial-market shocks
In the specifications considered so far, entrepreneurs and traders share the same valuation for
the installed capital. We now develop a variant of the model in which entrepreneurs and traders
have different valuations. In this variant, additional non-fundamental volatility originates from
correlated errors in the entrepreneurs’ expectations about the traders’ valuations; once again, our
mechanism amplifies the impact of these errors. This variant thus helps connect our model to the
recent work on speculative trading à la Harrison and Kreps (1978).31
We consider the following modification of the baseline model. The traders’ utility in period
t = 3 is given by (θ + ω) ki , where ω is a random variable, independent of θ and of any other
exogenous random variable in the economy, Normally distributed with mean zero and variance
σω2 . This random variable is a private-value component in the traders’ valuation. It can originate
from the hedging motive of the traders, from a different discount factor, or from heterogeneous
valuations à la Harrison and Kreps (1978). For our purposes, what matters is that the presence of
ω in the traders’ utility is taken as given by the social planner; that is, the planner respects the
preference orderings revealed by the agents’ trading decisions. We thus choose a neutral label for
ω and simply call it a “financial market shock.”
We also modify the entrepreneurs’ information set, to allow for information regarding ω to affect
investment decisions. In particular, the entrepreneurs observe a common signal w = ω+ζ, where ζ is
31
See Scheinkman and Xiong (2003), Gilchrist, Himmelberg, and Huberman (2005), and Panageas (2005).
30
common noise, independent of any other exogenous random variable in the economy, with variance
σζ2 . The signal w is observed by the entrepreneurs but not by the traders; as in the baseline model,
this is a shortcut for introducing correlated errors in the entrepreneurs’ expectations regarding the
financial-market shock. Finally, to focus on common expectational shocks about ω rather than
about θ, we remove the common signal y : the entrepreneurs observe only private signals about θ,
xi = θ + ξi , where ξi is idiosyncratic noise as in the baseline model.
In this environment, the asset price in period two is given by
p = E [θ|K, ω] + ω.
It follows that equilibrium investment choices depend not only on the entrepreneurs’ expectations
of θ, but also on their expectations of ω : there exist coefficients (β0 , β1 , β2 ) such that individual
investment is given by k (x, w) = β0 + β1 x + β2 w and, by implication, aggregate investment is given
by K (θ, w) = β0 +β1 θ +β2 w. Following similar steps as in the baseline model, leads to the following
result.
Proposition 10 (i) In any equilibrium, there exist a scalar α > 0 and a function κ(θ, ω) such
that
k (x, w) = E [ (1 − α) κ (θ, ω) + αK (θ, w) | x, w] .
(ii) λ small enough suffices for the equilibrium to be unique and for investment to increase with
both θ and w.
(iii) The efficient investment satisfies
k (x, w) = E [ θ + λω | x, w] .
(iv) In any equilibrium in which investment increases with both θ and w, investment underreacts
to θ and overreacts to w.
In this economy, entrepreneurs pay too much attention to their signals regarding shocks in the
financial market. The reason is essentially the same as in the benchmark model. When traders
interpret high investment as good news for θ, financial prices increase with aggregate investment.
Because the noise in the entrepreneurs’ signals about the financial market shock ω is correlated,
these signals are relatively better predictors of aggregate investment than the signals about θ. By
implication, entrepreneurs’ investment decisions are oversensitive to information about financial
market shocks relative to information about their fundamental valuation θ. Through this channel,
an increase in investment that was purely driven by expectations regarding financial market shock
is amplified.
31
Absent informational frictions (i.e., if θ were known at the time of financial trade)), the response
of investment to θ and ω would be efficient. Since ω can be interpreted as the difference between the
traders’ and the entrepreneurs’ fundamental valuations of the asset, this case is reminiscent of the
efficiency results obtained in richer models of “bubbles” based on heterogeneous priors; in particular, Panageas (2006) derives a similar efficiency result for a model that introduces heterogeneous
valuations in a q-theory model of investment. The interesting novelty here is that inefficiency arises
once we introduce dispersed information. Traders are then uncertain whether high investment is
driven by good fundamentals or by the entrepreneurs’ expectations of speculative valuations. This
uncertainty opens the door to our feedback effect between financial prices and investment, creating
inefficiency in the response of investment to different sources of information.
6.4
Other extensions
An important function of stock prices is to guide corporate investment choices by revealing valuable
information that is dispersed in the marketplace and not directly available to corporate managers
(e.g., Dow and Gorton, 1997; Subrahmanyam and Titman, 1999; Chen, Goldstein and Jiang, 2007).
This effect is absent in the preceding analysis, because the entrepreneurs’ investment choices are
made before the opening of the financial market. However, we can easily incorporate such an effect
by letting the entrepreneurs make an additional investment in stage two, after observing the price
in the financial market.32 Provided that the dispersion of information does not vanish, the source of
complementarity and inefficiency we have documented remains. Interestingly, though, an additional
information externality emerges: if all agents were to increase their reliance on idiosyncratic sources
of information, then the information contained in prices would be more precise, which in turn would
improve the efficiency of the investment decisions that follow the observation of these prices. Clearly,
this informational externality only reinforces the conclusion that agents rely too much to common
sources of information, and hence that non-fundamental volatility is inefficiently high.
Throughout the preceding extensions, we have maintained the assumption that traders cannot
directly invest in the new technology during the first period. Clearly, our results do not hinge
on this assumption. For example, consider the benchmark model and suppose that each trader
j chooses first-period real investment kj at cost kj2 /2 and then trades an additional qj units in
the second-period financial market. Neither the equilibrium price in the financial market nor the
entrepreneurs’ choices in the first period are affected; all that happens is that aggregate investment
now includes the investment of the traders, which is simply given by kT = Eθ, which does not affect
the information structure in the second period. More generally, one could drop the distinction
32
Alternatively, we could introduce a financial market in stage 1 or let entrepreneurs observe a noisy signal of K
instead of a noisy price signal.
32
between entrepreneurs and traders altogether and simply talk about differentially informed agents
who first make real investment decisions and then trade financial claims on the installed capital.
Next, consider the assumption that a fraction λ of the entrepreneurs is hit by a liquidity shock
and is forced to sell their capital in the financial market; this was a modeling device that ensured that
the private return to first-period investment depends on (anticipated) second-period financial prices
while ensuring tractability. If one were to drop the assumption of risk neutrality, or assume that
the second-period transaction costs depend on gross positions, or introduce short-sale constraints
in the financial market, then the profits an agent could do in the financial market would depend
on how much capital he enters the market with; this in turn would ensure that private returns to
first-period investment depend on expectations of future financial prices, even in the absence of
liquidity shocks.33
Finally, consider the assumption that profitability is perfectly correlated across entrepreneurs.
Clearly, what is essential is only that there is a common component about which agents have
dispersed information. For example, we could let the productivity of the new technology for entrepreneur i be θ̃i = θ +vi , where θ is the common component and vi is an idiosyncratic component;
we could then also let the entrepreneurs’ signals be θ̃i plus noise instead of θ plus noise. Alternatively, we could introduce common and idiosyncratic shocks to the entrepreneurs’ cost of investment
during period 1. In this case, unobservable common shocks to the cost of investment would also
act as a source of noise in the information that aggregate investment conveys about θ, essentially
playing the same role as the correlated errors in the entrepreneurs’ signals about θ.34
7
Conclusion
This paper examined the interaction between real and financial decisions in an economy in which
information about underlying profitability is dispersed. By conveying a positive signal about profitability, higher aggregate investment stimulates higher asset prices, which in turn raise the incentives to invest. This creates an endogenous complementarity, making investment decisions sensitive
to higher-order expectations. In turn, this can dampen the impact of fundamental shocks and
amplify the impact of common expectational shocks. Importantly, all these effects are symptoms
of inefficiency.
These effects are likely to be stronger during periods of intense technological change, when the
dispersion of information about the potential of the new technologies is particularly high. Our analysis therefore predicts that such periods come hand-in-hand with episodes of high non-fundamental
33
Note, however, that these extensions may feature additional deviations from the first best (e.g., short-sale constraints), which may introduce novel effects in addition to the ones we have considered.
34
Such an extension is studied in the Supplementary Material.
33
volatility and comovement in investment and asset prices. At some level, this seems consistent with
the recent experiences surrounding the internet revolution or the explosion of investment opportunities in China. What looks like irrational exuberance may actually be the amplified, but rational,
response to noise in information. While both explanations open the door to policy intervention,
the one suggested by our theory is not based on any presumption of “intelligence superiority” on
the government’s side.
Our mechanism also represents a likely source of non-fundamental volatility and inefficiency over
the business cycle. Indeed, information regarding aggregate supply and demand conditions seems to
be widely dispersed in the population, which explains the financial markets’ anxiety preceding the
release of key macroeconomic statistics. Extending the analysis to richer business-cycle frameworks
is an important direction for future research.
34
Appendix: Proofs omitted in the main text
Proof of Lemma 3. The derivations of β1 and β2 are in the proof of the Lemma 4. Rearranging
(30), gives
β2 =
1
(1 − λ + λγ1 β1 ) δ2 .
1 − λγ1
Using (29), α = λγ1 and δ2 /δ1 = πy /πx , then gives the result.
Proof of Lemma 4.
The proof proceeds in several steps. We start by proving part (i). We
continue with some auxiliary results regarding the function F which are used in the last steps. We
conclude by establishing parts (ii), (iii) and (iv). Throughout, to simplify notation, we suppress
the dependence of F and G on (λ, πθ , πx , πy ) and let π ≡ πθ + πx + πy , δ0 ≡ πθ /π, δ1 ≡ πx /π, and
δ2 ≡ πy /π.
Part (i). Substituting K(θ, y) = β0 + β1 θ + β2 y into (7) and using E [θ|x, y] = δ0 µ + δ1 x + δ2 y
gives
k (x, y) = (1 − λ) E [θ|x, y] + λγ0 + λγ1 (β0 + β1 E [θ|x, y] + β2 y)
= (1 − λ + λγ1 β1 ) E [θ|x, y] + λγ0 + λγ1 β0 + λγ1 β2 y
= [(1 − λ + λγ1 β1 ) δ0 µ + λγ0 + λγ1 β0 ] +
+ [(1 − λ + λγ1 β1 ) δ1 ] x + [(1 − λ + λγ1 β1 ) δ2 + λγ1 β2 ] y
Because in equilibrium the above must coincide with β0 + β1 x + β2 y for all x and y, the following
conditions must hold
β0 = (1 − λ + λγ1 β1 ) δ0 µ + λγ0 + λγ1 β0 ,
(28)
β1 = (1 − λ + λγ1 β1 ) δ1 ,
(29)
β2 = (1 − λ + λγ1 β1 ) δ2 + λγ1 β2 .
(30)
It is immediate that any equilibrium must satisfy β1 6= 0. Then let b ≡ β2 /β1 . From (4) and (6),
γ1 β1 = h (b) ≡
δ2 (1 + b)
δ0 b2 + δ2 (1 + b)2
(31)
while from (29) and (30),
b=
δ2
λγ1 β1 b
+
.
δ1 (1 − λ + λγ1 β1 ) δ1
35
(32)
Substituting (31) into (32) gives b = F (b), where
δ2
F (b) ≡
δ1
1+
λ (1 + b) b
(1 − λ) (δ0 + δ2 ) b2 + (2 − λ)δ2 b + δ2
.
(33)
Note that the domain of F is the set of all b ∈ R such that 1 − λ + λγ1 β1 6= 0. Using (31), the
latter is given by
B ≡ {b ∈ R : (1 − λ)(δ0 + δ2 )b2 + (2 − λ)δ2 b + δ2 6= 0}.
It follows that, in any linear equilibrium, b is necessarily a fixed point of F , while the coefficients
(β0 , β1 , β2 , γ0 , γ1 ) are given by the following conditions:
β1 = [1 − λ + λh (b)]δ1
(34)
β2 = bβ1 = b[1 − λ + λh (b)]δ1
(35)
γ1 =
γ1 β1
h (b)
=
β1
[1 − λ + λh (b)]δ1
β0 = (1 − λ + λh (b)) δ0 µ +
γ0 =
(36)
λδ0 µ
δ0 + δ2 1 +
δ0
µ − γ1 β0
δ0 + δ2 (1 + 1b )2
(37)
1 2
b
(38)
Conditions (34)-(38) uniquely define the function G.
Auxiliary results. Let g(b) ≡ (1 − λ)(δ0 + δ2 )b2 + (2 − λ)δ2 b + δ2 ; the domain of F is B = {b ∈
R : g(b) 6= 0} and its complement is Bc = {b ∈ R : g(b) = 0}. Note that the discriminant of g(b) is
n
o
(2−λ)δ2
∆ ≡ (δ2 λ)2 − 4δ0 δ2 (1 − λ). If ∆ < 0, then Bc = ∅; if ∆ = 0, then Bc = − 2(1−λ)(δ
; finally,
0 +δ2 )
n
√
√ o
(2−λ)δ2 + ∆
(2−λ)δ2 − ∆
if ∆ > 0, then Bc = − 2(1−λ)(δ
, − 2(1−λ)(δ
. Because there are values for (δ0 , δ2 , λ) that
0 +δ2 )
0 +δ2 )
make ∆ negative, zero, or positive, all three cases are possible in general. However, because ∆ is
continuous in λ and ∆ = −4δ0 δ2 < 0 when λ = 0, λ small enough suffices for Bc = ∅. Moreover,
because g(b) ≥ δ2 > 0 for any b ≥ 0, R+ ⊂ B always.
The function F is continuously differentiable over its entire domain, with
F ′ (b) = λ
δ2 φ1 (b)
δ1 g (b)2
where φ1 (b) ≡ [δ2 − (1 − λ) δ0 ]b2 + 2δ2 b + δ2 . Moreover,
lim F (b) = lim F (b) = F∞
b→−∞
b→+∞
δ2
≡
δ1
and F (−1) = F (0) = δ2 /δ1 < F (δ2 /δ1 ) .
36
1+
λ
(1 − λ) (δ0 + δ2 )
>
δ2
,
δ1
Consider the case δ2 = (1 − λ) δ0 . Then φ1 (b) = 0 admits a unique solution at b = −1/2.
Because ∆ < 0, the function F is defined over the entire real line, it is decreasing for b < −1/2
and increasing for b > −1/2. Next, consider the alternative case, δ2 6= (1 − λ) δ0 . Then φ1 (b) = 0
admits exactly two solutions, at b = b1 and at b = b2 , where
p
p
−δ2 − (1 − λ)δ0 δ2
−δ2 + (1 − λ)δ0 δ2
b1 ≡
and b2 ≡
.
δ2 − (1 − λ) δ0
δ2 − (1 − λ) δ0
The function F then reaches a local maximum at b1 and a local minimum at b2 .
Part (ii). By the preceding results we have that F is continuous over R+ , with F (δ2 /δ1 ) > δ2 /δ1
and limb→∞ F (b) < ∞. It follows that the equation F (b) = b admits at least one solution at
b > δ2 /δ1 = πy /πx .
Part (iii). Fix any (δ1 , δ2 ) ∈ (0, 1)2 . If λ is such that δ2 = (1 − λ) δ0 , where δ0 = 1 − (δ1 + δ2 ),
then let F ≡ F (−1/2) and F̄ ≡ F∞ . If, instead, λ is such that δ2 6= (1 − λ) δ0 , then let F ≡
min{F∞ , F (b2 )} and F̄ ≡ max{F∞ , F (b1 )}. It is easy to check that both F and F̄ converge to
δ2 /δ1 as λ → 0. Since F is continuous over its entire domain, B, and λ small enough suffices for
B = R, we have that λ small enough also suffices for F to be bounded in [F , F̄ ]. But then F
converges uniformly to δ2 /δ1 as λ → 0. It follows that for any ε > 0 there exists λ̂ = λ̂ (ε) > 0 such
that, whenever λ < λ̂, B = R and F has no fixed point outside the interval [δ2 /δ1 − ε, δ2 /δ1 + ε].
Now note that the function F ′ (b; λ) is continuous at (δ2 /δ1 , 0) with F ′ (δ2 /δ1 ; 0) = 0. It follows
that, for any η ∈ (0, 1), there exist ε̃ = ε̃ (η) > 0 and λ̃ = λ̃ (η) such that −1 < −η < F ′ (b; λ) <
η < 1 for all b ∈ [δ2 /δ1 − ε̃, δ2 /δ1 + ε̃] and all λ ∈ [0, λ̃].
Combining the aforementioned results with the continuity of F , we have that there exist ε̄ > 0
and λ̄ > 0 such that, for any λ ∈ [0, λ̄], the following are true: for any b ∈
/ [δ2 /δ1 − ε̄, δ2 /δ1 + ε̄],
F (b) 6= b; for b ∈ [δ2 /δ1 − ε̄, δ2 /δ1 + ε̄], F is continuous and differentiable in b, with F ′ (b) < 1. It
follows that, for λ < λ̄, F has at most one fixed point. Together with the fact that F necessarily
has at least one fixed point (from part (ii)), this proves part (iii).
Part (iv). It is easy to check that (δ1 , δ2 , λ) = (.2, .1, .75) implies that B = R (so that F
is continuous over the entire real line) and F (b2 ) < b2 < 0. These properties, together with the
properties that F (0) > 0 and limb→−∞ F (b) > 0 > −∞, ensure that, in addition to a fixed point in
(δ2 /δ1 , +∞), F admits at least one fixed point in (−∞, b2 ) and one in (b2 , 0). Indeed, in this example
F admits exactly three fixed point, which are “strict” in the sense that F (b)−b changes sign around
them. Because F is continuous in (b, δ1 , δ2 , λ) in an open neighborhood of (δ1 , δ2 , λ) = (.2, .1, .75) ,
there necessarily exists an open set S ⊂ (0, 1)3 such that F admits three fixed points whenever
(δ1 , δ2 , λ) ∈ S.
Proof of Proposition 1.
From Lemma 4, there always equilibrium in which b > πy /πx . Pick
37
the equilibrium that corresponds to the highest solution to F (b; λ) = b and let b(λ) ∈ (πy /πx , +∞)
denote this solution. Part (i) follows from conditions (34), (35) and (36) observing that b > 0
suffices for h (b) > 0 and hence for β1 , β2 , γ1 > 0. For part (ii), note that α > 0 follows from γ1 > 0;
that α < 1 follows from (8) along with β2 /β1 > 0; finally, that α increases with λ follows from
Lemma 3 and part (iii), which we prove next. First, note that
∂F (b; λ)
δ2 b (1 + b) δ2 (1 + 2b) + (δ0 + δ2 ) b2
=
,
∂λ
δ1
g (b)2
so that b > 0 suffices for ∂F (b; λ) /∂λ > 0. Next, note that the function F (b; λ) − b is continuous
in b over (πy /πx , +∞) and satisfies limb→+∞ {F (b; λ) − b} = −∞. Since b(λ) ∈ (πy /πx , +∞) is
the highest solution to F (b; λ) − b = 0, it is then necessarily the case that F (b; λ) − b < 0 for any
b > b(λ). Part (iii), then follows from this property together with the fact that F increases with λ.
Proof of Proposition 2. Take any λ < λ̄. Let b (λ) denote the unique fixed point to F (b; λ) = b
and denote by β0 (λ) , β1 (λ) , β2 (λ) , γ0 (λ) and γ1 (λ) the corresponding equilibrium coefficients,
as given by (34)-(38). Note that all these functions are continuous.
Part (i). Using conditions (34)–(38), the sensitivity of investment to the realization of θ is given
by
β1 (λ) + β2 (λ) = W (λ) (δ1 + δ2 ) ,
1
where W (λ) ≡ w (b (λ) , λ) , with w (b, λ) ≡ (1 + b) (1 − λ + λh (b)) δ1δ+δ
and h (b) defined as in
2
(31). We can compute b′ (λ) and b′′ (λ) applying the Implicit Function Theorem to F (b, λ) − b. We
can then use this to compute W ′ (λ) and W ′′ (λ) . After some tedious algebra (which is available
upon request), we find that W ′ (0) = 0 and W ′′ (0) = − (δ
2δ0 δ1 δ2
1 (δ1 +δ2 )+δ2 )
2
< 0. Together with the fact
that b (0) = δ2 /δ1 and hence W (0) = 1, this ensures that there exists λ̂ ∈ (0, λ̄] such that, for all
λ ∈ (0, λ̂), W (λ) < W (0) = 1 and W ′ (λ) < 0; that is, β1 + β2 is lower than δ1 + δ2 , its value in
the frictionless benchmark, and is decreasing in λ.
Part (ii). From condition (34), we have that β1 (λ) = [1 − λ + λh (b (λ))]δ1 and hence β1′ (λ) =
δ1 [−1 + h (b (λ)) + λh′ (b (λ) b′ (λ))] . Since b (0) = δ2 /δ1 and h (δ2 /δ1 ) =
that
β1′ (0)
δ1 (δ1 +δ2 )
δ1 (δ1 +δ2 )+δ2
< 1, we have
= δ1 [−1 + h (δ2 /δ1 )] < 0, which together with the result from part (i) that β1′ (0) +
β2′ (0) = 0 gives β2′ (0) > 0. The result then follows from the local continuity of β2′ (λ) in λ.
Proof of Proposition 4. The first claim is proved by the numerical example in the main text.
Thus consider the second claim. Given any linear strategy k(x, y) = β0 + β1 x + β2 y, ex-ante utility
38
is given by
1
2
2Eu = E − k (x, y) + θk (x, y)
2
1
1
1
= − β02 + β0 (1 − β1 − β2 ) µ − β12 πx−1 − (β1 + β2 )2 πθ−1
2
2
2
1 2 −1
1
2
− β2 πy + (β1 + β2 ) σθ + (β1 + β2 ) 1 − (β1 + β2 ) µ2 .
2
2
(39)
Now suppose prices are fully stabilized at p = p̄. Substituting p(θ, y) = p̄ into the entrepreneurs’
best response (2) gives the following coefficients for the equilibrium investment strategy:
β0 = (1 − λ) δ0 µ + λp̄,
β1 = (1 − λ) δ1 ,
and β2 = (1 − λ) δ2 .
(40)
Note that p̄ affects only the first two terms in (40) through its effect on β0 . Hence, the maximal
welfare that can be achieved with full price stabilization is obtained by choosing p̄ so that β0 =
1 − (1 − λ)(δ1 + δ2 ).
Next, note that for any a ∈ (0, 1) and any b ∈ R, there exists a policy τ (p) = τ0 + τ1 p that
induces an equilibrium in which the investment strategy is given by35
β0 = b
β1 =
(1 − λ)δ1
1 − aδ1
and
β2 =
(1 − λ + aβ1 )δ2
.
1−a
(41)
To see this, suppose that, given (τ0 , τ1 ), the entrepreneurs follow the linear strategy defined in (41).
Then E[θ|K] = γ0 + γ1 K, where (γ0 , γ1 ) are obtained from (41) using the formulas given in (6).
The market clearing price is then equal to
p=
1
(γ0 + γ1 K − τ0 )
1 + τ1
(42)
Replacing (42) and K(θ, y) = β0 + β1 θ + β2 y into (2), we then have that the best response for each
entrepreneur consists in following the strategy k (x, y) = β̃0 + β̃1 x + β̃2 y given by
λ(γ0 − τ0 )
+ (1 − λ) δ0 µ + α̃ [β0 + β1 δ0 µ]
1 + τ1
= (1 − λ) δ1 + α̃β1 δ1
β̃0 =
β̃1
β̃2 = (1 − λ) δ2 + α̃ [β1 δ2 + β2 ]
where α̃ = λγ1 /(1 + τ1 ). It is then immediate that there exists a (τ0 , τ1 ) such that β̃0 = β0 , β̃1 = β1
35
Equivalently, for any a ∈ (0, 1) and any κ0 ∈ R, there exists a policy (τ0 , τ1 ) that sustains
an equilibrium in which
the investment strategy satisfies k(x, y) = E[(1 − a)κ̃(θ) + aK(θ, y) | x, y] with κ̃(θ) ≡
39
1−λ
1−a
θ + κ0 .
and β̃2 = β2 (it suffices to choose τ1 so that α̃ = a and then adjust τ0 so that β̃0 = b).
Now let b0 ≡ 1 − (1 − λ)(δ1 + δ2 ) and for any a ∈ [0, 1) let
β̃1 (a) ≡
(1 − λ) δ1
,
1 − aδ1
β̃2 (a) ≡
(1 − λ + aβ1 )δ2
,
1−a
and
h
i
2
2
1
1
1
β̃1 (a) πx−1 −
β̃1 (a) + β̃2 (a) πθ−1
W (a) ≡ − b20 + b0 1 − β̃1 (a) − β̃2 (a) µ −
2
2
2
2
1
−1
−1
− β̃2 (a) πy + β̃1 (a) + β̃2 (a) πθ
2
1
+ β̃1 (a) + β̃2 (a) 1 −
β̃1 (a) + β̃2 (a) µ2
2
Note that welfare under full price stabilization is given by W (0), whereas welfare under any policy
(τ0 , τ1 ) that implements a linear strategy as in (41) with a ∈ (0, 1) and b = b0 is given by W (a). Next
note that W is continuously differentiable over [0, 1). To prove the second claim in the proposition
it thus suffices to show that
dW
∂W dβ̃1
∂W dβ̃1
=
+
>0
da
da
∂ β̃1
∂ β̃2 da
at a = 0. First note that
∂W
∂ β̃1
∂W
∂β2
h
i
= −b0 µ − β̃1 πx−1 − β̃1 + β̃2 πθ−1 + πθ−1 + 1 − β̃1 + β̃2 µ2
h
i
= −b0 µ − β̃2 πy−1 − β̃1 + β̃2 πθ−1 + πθ−1 + 1 − β̃1 + β̃2 µ2
Using β̃1 (0) = (1 − λ) δ1 , β̃2 (0) = (1 − λ) δ2 and b0 ≡ 1 − (1 − λ)(δ1 + δ2 ), we thus have that
∂W
∂ β̃1
=
β̃1 =β̃1 (0)
∂W
∂ β̃2
= −b0 µ + [1 − (1 − λ) (δ1 + δ2 )] µ2 + λπθ−1 = λπθ−1 > 0.
β̃2 =β̃2 (0)
Because β̃1 and β̃2 are both increasing in a, it follows that dW (0)/da is positive, which establishes
the result.
Proof of Proposition 5. Rewrite the tax rule as
τ (θ, K) = −φ0 − (φ1 − 1)θ − φ2 K,
where φ0 ≡ −τ0 , φ1 ≡ 1−τ1 , and φ2 ≡ −τ2 . Suppose that all other entrepreneurs follow the efficient
40
strategy k(x, y) = E [θ|x, y]. The equilibrium price is then given by
p(θ, y) = E [θ − τ (θ, K(θ, y)) | K(θ, y)] = φ0 + φ1 E [θ | K(θ, y)] + φ2 K(θ, y)
= φ0 + φ1 γ0 + (φ1 γ1 + φ2 ) K(θ, y)
where we have used E [θ | K(θ, y)] = γ0 + γ1 K(θ, y), with (γ0 , γ1 ) determined by substituting
(β0 , β1 , β2 ) = (δ0 µ, δ1 , δ2 ) into (6). The best response for each individual entrepreneur is then to
follow the strategy
k(x, y) = E [(1 − λ) (θ − τ ) + λp | x, y]
(43)
= E [(1 − λ) (φ0 + φ1 θ + φ2 K) + λ (φ0 + φ1 γ0 + (φ1 γ1 + φ2 ) K) | x, y]
= (1 − λ) φ0 + λ (φ0 + φ1 γ0 ) + (1 − λ) φ1 E [θ|x, y] + [(1 − λ) φ2 + λ (φ1 γ1 + φ2 )] E [K|x, y]
For the tax τ (θ, K) to implement the efficient allocation, it is thus necessary and sufficient that the
strategy in (43) coincides with k(x, y) = E [θ|x, y] , which is possible if and only if
(1 − λ) φ2 + λ (φ1 γ1 + φ2 ) = 0,
(1 − λ) φ1 = 1,
and
(1 − λ) φ0 + λ (φ0 + φ1 γ0 ) = 0.
Equivalently,
τ0 = −φ0 =
λ
γ0 ,
1−λ
τ1 = 1 − φ1 = −
λ
,
1−λ
and τ2 = −φ2 =
λ
γ1 .
1−λ
The result then follows from the fact that γ0 < 0 < γ1 when the entrepreneurs follow the efficient
allocation.
Proof of Proposition 6.
For part (i), it suffices to substitute the price as in (16) into the
entrepreneurs’ best response (2). Thus consider part (ii). Substituting (4) into (6) gives
γ1 =
πz
1
(β1 + β2 ) πy
(β1 + β2 ) δ2
= 2
= 2
.
2
πθ + πz β1 + β2
β2 πθ + (β1 + β2 ) πy
β2 δ0 + (β1 + β2 )2 δ2
In the limit, as λ → 0, we have that β0 → δ0 , β1 → δ1 , β2 → δ2 , and hence γ1 →
(δ1 +δ1 )δ2
δ22 δ0 +(δ1 +δ1 )2 δ2
> 0.
By continuity, then, there exists λ̂ > 0 such that, for all λ ∈ (0, λ̂), (β1 + β2 ) > 0, i.e. investment
increases with θ, γ1 > 0, i.e. the traders’ expectation of θ increases with K, and α = λ(γ1 −λ/φ) > 0,
i.e. entrepreneurs perceive a complementarity in their investment decisions.
Proof of Proposition 7. Let V (k, K, θ) ≡ − 12 k2 + θk −
λ2 2
2φ K .
From (19), Eu = 12 EV (k, K, θ).
The result then follows from Proposition 3 in Angeletos and Pavan (2007a), noting that κ∗ (θ) ≡
41
arg maxK V (K, K, θ) =
1
1+λ2 /φ θ
Proof of Proposition 8.
and α∗ ≡ 1 −
Vkk +2VkK +VKK
Vkk
= VKK = −λ2 /φ.
From (2), in any equilibrium in which p is linear in (θ, y, ω), there
are coefficients (β0 , β1 , β2 ) such that k(x, y) = β0 + β1 x + β2 y. From (22) and (23), the equilibrium
price is then
p(θ, y, ω) = P (K(θ, y), θ, ω) ≡ η0 + η1 K(θ, y) + η2 θ + η3 ω.
(44)
for some (η0 , η1 , η2 , η3 ).
Now consider the optimality of the traders’ strategies. As in the benchmark model, the information that K(θ, y) reveals about θ is the same as that of a signal
z≡
whose precision is πz ≡
β1 +β2
β2
2
K(θ, y) − β0
β2
=θ+
ε
β1 + β2
β1 + β2
πy ,while the information that p(θ, y, ω) reveals about θ given
K(θ, y) is the same as that of a signal
s=
whose precision is πs =
distributed with mean
2
η2
η3
1
η3
[p(θ, y, ω) − η0 − η1 K(θ, y)] = θ + ω
η2
η2
πω . A trader who observes K and p thus believes that θ is normally
πθ
πz
πs
µθ +
z+
s
πθ + πz + πs
πθ + πz + πs
πθ + πz + πs
= γ0 + γ1 K(θ, y) + γ2 θ + γ3 ω
E [θ | K(θ, y), p(θ, y, ω)] =
where
πθ
πz
β0
µθ −
πθ + πz + πs
πθ + πz + πs β1 + β2
1
πz
γ1 =
πθ + πz + πs β1 + β2
πs
γ2 =
πθ + πz + πs
πs
η3
γ3 =
.
πθ + πz + πs η2
γ0 =
42
(45)
(46)
(47)
(48)
Combining (22) with (44) we then have that
η0 =
η1 =
η2 =
η3 =
γ0
2−λ
1
λ
γ1 −
2−λ
φ
1
(γ2 + 1 − λ)
2−λ
1
(γ3 − 1 + λ) .
2−λ
(49)
(50)
(51)
(52)
Lastly, consider the optimality of the entrepreneurs’ investment strategies. From condition
(2), the strategy k (x, y) = β0 + β1 x + β2 is individually rational if and only if (β0, β1 , β2 ) satisfy
β0 +β1 x+β2 y = (1 − λ) E [θ|x, y]+λE [p(θ, y, ω)|x, y] . That is, (β0, β1 , β2 ) must satisfy the following:
β0 = [1 − λ + λη1 β1 + λη2 ]δ0 µθ + λη0 + λη1 β0
(53)
β1 = (1 − λ + λη1 β1 + λη2 ) δ1
(54)
β2 = (1 − λ + λη1 β1 + λη2 ) δ2 + λη1 β2
(55)
A linear equilibrium is a thus a solution to (45)-(55).
The existence of a linear equilibrium and its uniqueness for λ small enough can be established
following steps similar to those in the benchmark model. Here we prove that λ small enough suffices
for γ1 > 0, and even for α > 0.
2
2
η2
2
Substituting πz ≡ β1β+β
π
and
π
=
πω = πω into (46) gives
y
s
η3
2
γ1 =
=
=
πz
1
πθ + πz + πs β1 + β2
(β1 + β2 ) πy
β22 πθ + (β1 + β2 )2 πy + β22 πω
(β1 + β2 ) δ2
β22 δ0 + (β1 + β2 )2 δ2 +
β22 πω
πθ +πy +πx
.
In the limit, as λ → 0, we have that β0 → δ0 , β1 → δ1 , β2 → δ2 , and hence
γ1 →
(δ1 + δ1 ) δ2
δ22 δ0
+ (δ1 + δ1 )2 δ2 +
δ22 πω
πθ +πy +πx
> 0.
By continuity, then, there exists λ̂ > 0 such that, for all λ ∈ (0, λ̂), (β1 + β2 ) > 0, γ1 > 0 and
λ
α = 2−λ
γ1 − λφ > 0.
43
Proof of Proposition 9. Let
λ2
1
K 2.
V (k, K, θ) ≡ θk − k2 −
2
2φ(2 − λ)
The result then follows for the same argument as in the proof of Proposition 7.
Proof of Proposition 10. Part (i). In any equilibrium in which the price p (θ, ω, w) is linear in
(θ, ω, w), there are coefficients (β0 , β1 , β2 ) such that the investment strategy can be written as
k (x, w) = β0 + β1 x + β2 w,
implying that aggregate investment satisfies K(θ, w) = β0 + β1 θ + β2 ω + β2 ζ. For the traders, who
know ω but do not know either ζ or θ, observing K is then equivalent to observing a Gaussian
signal z with precision πz , where
β2
K − β0 − β2 ω
=θ+ ζ
z≡
β1
β1
and πz ≡
β1
β2
2
πζ ,
with πζ ≡ σζ−2 . It follows that the equilibrium price satisfies
p (θ, ω, w) = E[θ|K, ω] + ω = γ0 + γ1 K (θ, w) + (1 − γ1 β2 ) ω,
where
γ0 =
πθ
πz β0
πz
µ−
and γ1 =
=
πθ + πz
πθ + πz β1
β1 (πθ + πz )
Substituting (56) into the entrepreneurs’ best response gives
β1
πζ
.
2
β2
πθ + πζ
β1
(56)
(57)
k(x, w) = (1 − λ) E [θ|x, w] + λE [p (θ, ω, w) |x, w]
= (1 − λ) E [θ|x, w] + λγ0 + λγ1 E [K (θ, w) |x, w] + λ (1 − γ1 β2 ) E [ω|x, w]
which can be rewritten as in part (i) of the proposition by letting
α ≡ λγ1
and
κ (θ, ω) ≡
(1 − λ) θ + λγ0 + λ (1 − γ1 β2 ) ω
.
1 − λγ1
Finally, that α > 0 is shown in the next part.
Part (ii). Substituting K (θ, w) = β0 + β1 θ + β2 w into (58) gives
k(x, w) = λ (γ0 + γ1 β0 ) + (1 − λ + λγ1 β1 ) E [θ|x, w] + λγ1 β2 w + λ (1 − γ1 β2 ) E [ω|x, w]
44
(58)
Using the facts that E [θ|x, w] = E [θ|x] = δ0 + δ1 x and E [ω|x, w] = E [ω|w] = ηw, where δ0 ≡
σθ−2 /(σθ−2 + σx−2 )µ, δ1 ≡ σx−2 /(σθ−2 + σx−2 ), and η ≡ σζ−2 /(σω−2 + σζ−2 ), the above reduces to
k (x, w) = λ (γ0 + γ1 β0 ) + (1 − λ + λγ1 β1 ) δ0 + (1 − λ + λγ1 β1 ) δ1 x
+λ [η + (1 − η) γ1 β2 ] w
For this strategy to coincide with k (x, w) = β0 + β1 x + β2 w, it is necessary and sufficient that the
coefficients (β0 , β1 , β2 ) solve the following system:
β0 = λ (γ0 + γ1 β0 ) + (1 − λ + λγ1 β1 ) δ0
(59)
β1 = (1 − λ + λγ1 β1 ) δ1 ,
(60)
β2 = λ [η + (1 − η) γ1 β2 ] .
(61)
By (57),
γ1 β1 =
πζ
∈ (0, 1) ,
2
β2
π
+
π
θ
ζ
β1
(62)
which together with (60) guarantees that β1 ∈ (0, δ1 ). From (60) and (61) we then get
2
πθ + πζ + (1 − η) πζ ββ21
β2
=λ
2
β1
β2
δ1 (1 − λ)
πθ + πζ + λπζ
β1
η
β2
β1
or equivalently
β2
=F
λβ1
where
η
F (b; λ) ≡
δ1
1+λ
β2
;λ
λβ1
λ2 πθ b2 + πω b
(1 − λ) λ2 πθ b2 + πζ
.
It is then easy to show that, for λ small enough, F has a unique fixed point and this fixed point is
in a neighborhood of
β2
η
= .
λβ1
δ1
Along with the fact that β1 > 0 always, this guarantees that β2 > 0 for λ small enough.
Part (iii). The social planner’s problem can be set up as in the baseline model, giving the
optimality condition stated in part (iii) of the proposition.
Part (iv). From part (iii) the efficient strategy is given by
k (x, w) = β0∗ + β1∗ x + β2∗ w
45
with
β0∗ = δ0 ,
β1∗ = δ1
and
β2∗ = λη.
We have already shown, in the proof of part (ii), that β1 < δ1 = β1∗ , which means that investment
underreacts to θ. Next, note that β1 > 0 implies γ1 > 0. From (61) it then follows that, in any
equilibrium in which β2 > 0, it is also the case that
β2 = λη + λ (1 − η) γ1 β2 > λη = β2∗ ,
which means that investment overreacts to w.
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