By allowing the initial prior over market size to be a mixture of distributions, this paper extends the model of irreversible investment under uncertainty proposed by Rob (1991). We find that capacity expansion fuels investors' optimism. It is shown in the paper that the crash is always preceded by a boom when the initial prior is a mixture of exponential distributions.Learning Investment Uncertainty

Julien Prat

English

Research Papers in Economics

Increasing optimism and demand uncertainty 
Julien Prat
Department of Economics, Vienna University
Abstract
By allowing the initial prior over market size to be a mixture of distributions, this paper
extends the model of irreversible investment under uncertainty proposed by Rob (1991). We
find that capacity expansion fuels investors' optimism. It is shown in the paper that the crash
is always preceded by a boom when the initial prior is a mixture of exponential distributions.
I am most thankful to Boyan Jovanovic for his encouragement. I am also very grateful to Giuseppe Bertola for his guidance, and
to Carlos Alos−Ferrer, Christopher Flinn, Mohamad L. Hammour, Omar Licandro, Ludovic Renou and Emily Sinnot for their
very helpful comments.
Citation: Prat, Julien, (2005) "Increasing optimism and demand uncertainty." Economics Bulletin, Vol. 12, No. 10 pp. 1−8
Submitted: April 13, 2005.  Accepted: June 15, 2005.
URL: http://www.economicsbulletin.com/2005/volume12/EB−05L10014A.pdf1 Introduction
It is well known that anticipated changes in market fundamentals may produce asset
price paths that would appear similar to speculative bubbles. A classical example of
such changes is the “Peso problem” considered by Krasker (1980), where fundamentals
are subject to regime switches. Rob (1991) investigates the implications of the “Peso
problem” when investment is irreversible.
More recently, Barbarino and Jovanovic (2004) have elaborated on Rob’s (1991) model
in order to explain the crash of the NASDAQ index in March 2001. They ﬁnd that, for
the model to ﬁt the data, investors must become more and more optimistic about future
prospects as output grows. Such a requirement hardly pleads in favor of the model because
intuition suggests that the likelihood of saturating the market increases with installed
capacity. Moreover, most of the statistical distributions do not have a decreasing hazard
rate (Fudenberg and Tirole (1991), p.267). This is probably why increasing optimism is
often assimilated to irrational exuberance.
This paper questions the common wisdom by showing that increasing optimism arises
endogenously in Rob’s (1991) set-up when parameter uncertainty is introduced. Instead
of assuming that anticipations are based on a speciﬁc distribution, we consider the more
general hypothesis according to which the initial prior is a mixture of distributions. We
show that when this is the case, market expansion fuels investors optimism. Consequently,
we can relax the requirements needed to generate booms. We ﬁnd that when the initial
prior is a mixture of exponential distributions, the crash is always preceded by a run-up
in capacity building.
Section 2 lays out the set up of the model. Optimal accumulation paths are charac-
terized in Section 3. Section 4 concludes.
2S e t u p o f t h e m o d e l
2.1 The structure of the industry
(i) Consumers’ demand: We assume that consumers’ demand is given by a stationary
inverse demand function. As in Barbarino and Jovanovic (2004), we restrict our attention
to the rectangular-demand case in order to alleviate the algebra. Therefore
D(kt;z)=
½
p;i fkt ≤ z
0;i fkt >z
where p is the price paid for one unit of output and kt is the total output of the industry
at date t.T h ev a r i a b l ez is the maximum number of new consumers demanding one unit
of the good per unit of time. Therefore z ∈ R+. z does not change over time but it is
initially unknown. In the remainder of the paper, we will say that the market crashes
when kt exceeds z for the ﬁrst time. Since the willingness to pay of each consumer p is
known, the only uncertain variable is z.
1(ii) Production: Capital does not depreciate and investment is irreversible. A unit of
capital produces one unit of output and its salvage value is negligible. The free-entry
c o n d i t i o nh o l d sa n de a c he n t r a n ti n c u r saﬁxed cost c. Machines take one period to be
activated and are operated without costs.
2.2 The Investors’ problem
We study a new industry so that installed capacity at the initial date is equal to zero.
Investors think of z as a random variable drawn at time 0 from a prior distribution I(z).
The prior distribution is common knowledge among the potential entrants. Given the
lack of historical data, investors’ beliefs are quite vague. To capture their uncertainty,
we consider that I(z) is a mixture of distributions. As explained by De Groot (1970,
p.190-191), this assumption is more accurate when uncertainty is pervasive.
In order to keep the analysis tractable, we restrict our attention to the case where the
prior is formed over a conjugate family of distributions F(z|λ) that can be parametrized
using the scalar parameter λ. The initial prior might be a mixture of exponential distri-
butions or, for example, a mixture of log-normal distributions ranked according to their
location parameter. The probability measure P0(λ) assigns to each distribution a relative
weight, so the initial prior reads
I(x) ≡ Pr(z ≤ x|k0 =0 )=
Z λ
λ
F(x|λ)dP0(λ) (1)
where it is assumed that λ belongs to
£
λ,λ
¤
⊂ R+. The speciﬁcation used in Rob (1991)
is a particular case of (1), with P0(λ) equal to the Dirac’s delta function δ(λ − λ
∗) for a
given λ
∗.
One may think of each λ as indexing a speciﬁce v a l u a t i o no ft h em a r k e t .I ne m e r g i n g
industries, investors usually face diﬀerent estimates of their business prospects. When
making their investment decision, they have to weight these diﬀerent predictions: a process
that is captured by the probability measure P0(λ). As opposed to z which refers to an
objective constraint, λ is a subjective parameter that allows to analyze the process through
which agents revise their expectations.
As time goes by until z is reached, new ﬁrms enter the market and aggregate output
kt increases. All investors perfectly observe industry’s output. In light of this experience,
potential entrants revise their belief about z. Parameter uncertainty introduces a new
dimension into the learning process: agents do not solely update each distribution F(x|λ),
but they also revise Pt(λ) by reallocating the relative weights attached to each λ. Since
the only new information acquired during the periods preceding the crash is that z ≥
kt ≥ kt−1, Bayes’ rule yields
Pt(λ|kt)=
R λ
λ (1 − F(kt|λ))dP0(λ)
R λ
λ (1 − F(kt|λ))dP0(λ)
(2)
2Relation (2) follows from Bayes’ rule because (1 − F(kt|λ)) is the probability that
the economy will reach capacity kt without experiencing a crash. Intuitively, pessimistic
predictions about the size of the market become less and less credible as the industry
expands. This new dimension of the learning process generates increasing optimism. To
formalize the intuition, we have to order the distributions F(x|λ) a c c o r d i n gt ot h e i r“ o p -
timism”. We propose to consider the case where they can be ranked using the Monotone
Hazard Rate Condition (henceforth MHRC), so that
f(x|λ1)
1 − F(x|λ1)
<
f(x|λ2)
1 − F(x|λ2)
, for all x>0 and λ1 < λ2 (3)
where f(x|λ) is the density associated to F(x|λ). It is easy to verify that families of
exponential or Pareto distributions satisfy condition (3). A c c o r d i n gt ot h eM H R C , the
lower is λ, the smaller is the hazard rate of the crash for any given output. Using this
criterion, Proposition 1 proves that posteriors become more and more skewed towards low
values of λ as kt increases.
Proposition 1 Assume that the distributions F(x|λ) can be ranked according to the
MHRC deﬁned in (3). Then the prior Pt(λ|kt) First Order Stochastically Dominates the
posterior Pt+1(λ|kt+1).
Proof. Let pt(λ|kt) denote the conditional density of λ.F r o m(2), it follows that
µ
pt(λ1|kt)
pt(λ2|kt)
¶
/
µ
pt+1(λ1|kt+1)
pt+1(λ2|kt+1)
¶
=
µ
1 − F(kt|λ1)
1 − F(kt+1|λ1)
¶
/
µ
1 − F(kt|λ2)
1 − F(kt+1|λ2)
¶
= e
−
R kt+1
kt
h
f(x|λ2)
1−F(x|λ2)−
f(x|λ1)
1−F(x|λ1)
i
dx < 1
where λ1 > λ2. Hence Pt(λ|kt) ÂMLR Pt+1(λ|kt+1), where ÂMLR orders distributions
according to the Monotone Likelihood Ratio Condition (MLRC henceforth). Since the
MLRC implies First Order Stochastic Dominance, the claim follows.
Now that we have characterized the learning process and shown that parameter un-
certainty delivers increasing optimism, we proceed by deriving the optimal accumulation
paths.
3 The optimal accumulation path
After the crash, investment collapses to zero and output remains constant. So we restrict
our discussion to the only periods of interest, namely those during which kt ≤ z. At any
point in time before the crash, the value of an installed unit is made of current proﬁts
plus the expected value of the unit next period times the discount factor β
V (kt|z ≥ kt)=p + βE [V (kt+1|z ≥ kt)] (4)
3Given that machines take one period to be activated and that c i st h ec o s to fe n t e r i n g
the industry, the free-entry condition is satisﬁed if and only if
c = βE [V (kt+1|z ≥ kt)] (5)
Substituting from (5) into (4) and iterating one period ahead, yields
V (kt|z ≥ kt)=V (kt+1|z ≥ kt+1)=p + c (6)
Hence, the value of a unit of capital remains constant through time prior to the crash.
Using this result, we can decompose the expected value. Equation (6) and Bayes rule
yield
E [V (kt+1|z ≥ kt)] = V (kt+1|z ≥ kt+1)Pr{z ≥ kt+1|z ≥ kt}
=( p + c)
⎛
⎝
R λ
λ (1 − F(kt+1|λ))dPt(λ|kt)
R λ
λ (1 − F(kt|λ))dPt(λ|kt)
⎞
⎠ (7)
The ﬁrst equality holds because the salvage value of capital is negligible. The ﬁnal
expression is derived using Bayes rule to compute the likelihood of the crash given current
output. Finally, substituting (6) and (7) into (4), we obtain
c
β (p + c)
=
R λ
λ (1 − F(kt+1|λ))dPt(λ|kt)
R λ
λ (1 − F(kt|λ))dPt(λ|kt)
(8)
Equation (8) states that the likelihood of the crash occurring next period remains
constant. Since the right hand side of (8) is a probability, the industry is viable if and
only if β >
¡
1+
p
c
¢−1 . So investors have to be patient enough.
Barbarino and Jovanovic (2004) analyze in details the accumulation paths under the
speciﬁc assumption according to which Pt (λ|kt) is given by the Dirac’s delta function for
ag i v e nλ
∗.1 They show that the hazard rate of the prior distribution F(x|λ
∗) is the key
statistic. For investment to be increasing from one period to the other, the hazard rate
of the prior must be a decreasing function of output. Conversely, when the hazard rate is
increasing, optimal investment decreases. Obviously, when the hazard rate of F(x|λ
∗) is
constant, investment remains even through time.
Once parameter uncertainty is introduced, this simple result does not hold anymore.
As discussed in section 2.2., when output increases, posteriors attach more and more
weight to optimistic opinions about z.This new mechanism drives a positive and increasing
wedge between the optimal rate of investment in our set-up and the one obtained in the
absence of parameter uncertainty. To illustrate this fact, Proposition 2 analyses the
threshold case where the prior is a mixture of distributions with a constant hazard rate.
As opposed to the single prior case, Proposition 2 shows that the accumulation path is
always convex.
1However, they also consider a more general set-up where c is not anymore constant but is an increasing
function of the aggregate rate of investment.
4Proposition 2 The optimal rate of investment is increasing through time if:
1. The distributions F(k|λ) have a constant hazard rate.
2. The prior distribution P(λ) is non-degenerate and absolutely continuous.
Proof. Since the hazard rate of F(x|λ) is constant, we can assume without loss of
generality that F(x|λ)=1− e−λx. Inserting this expression in (2) and (8) yields the
following optimality condition
Z λ
λ
µ
e
−λkt+1 −
µ
c
β (p + c)
¶
e
−λkt
¶
e
−λktp0(λ)dλ =0 (9)
where p0(λ) denotes the density associated to P0(λ). The proof proceeds by contradiction:
(i) Let ∆ = ∆kt = kt+1 −kt denote aggregate investment in period t. Assume that invest-
ment remains constant in the next period, so that ∆kt+1 = ∆. By iteration, we obtain
g(s|kt)=
Z λ
λ
e
−2λ(kt+∆(s−t))
µ
e
−λ∆ −
µ
c
β (p + c)
¶¶
p0(λ)dλ (10)
=
Z λ
λ
h(λ|s,kt)dλ =0 ; for s = {t,t +1 }. (11)
where h(λ|s,kt) is given by
h(λ|s,kt)=e
−2λ(kt+∆(s−t))
µ
e
−λ∆ −
µ
c
β (p + c)
¶¶
p0(λ)
According to condition (10),g 0(t|kt) must be equal to zero. However
dg(s|kt)
ds
= −2∆
Z λ
λ
λh(λ|s,kt)dλ
= −2∆
"
λH
¡
λ
¯ ¯s,kt
¢
− λH (λ|s,kt) −
Z λ
λ
H (λ|s,kt)dλ
#
where H (λ|s,kt) is the primitive of h(λ|s,kt). For the optimality condition (10) to
be satisﬁed, the function h(λ|s,kt)|s=t must change sign in
£
λ,λ
¤
. But h(λ|s,kt) is
decreasing in λ. Hence the primitive H (λ|s,kt)|s=t reaches its minimum at λ, so that
Z λ
λ
H (λ|t,kt)dλ > (λ − λ)H
¡
λ
¯ ¯t,kt
¢
Equation (10) also implies that H
¡
λ
¯ ¯t,kt
¢
= H (λ|t,kt). Therefore
dg(s|kt)
ds
¯ ¯ ¯ ¯
s=t
= −2∆
"
(λ − λ)H
¡
λ
¯ ¯t,kt
¢
−
Z λ
λ
H (λ|t,kt)dλ
#
> 0
5which contradicts the optimality condition at date t +1 .
(ii) Let’s assume that ∆kt > ∆kt+1. At date t, condition (9) still holds. Iterating one
period ahead, the optimality condition at date t +1reads
Z λ
λ
e
−2λ(kt+∆kt)
µ
e
−λ∆kt+1 −
µ
c
β (p + c)
¶¶
p0(λ)dλ >g (t +1 |kt) > 0
The only case which does not lead to a contradiction is when ∆kt < ∆kt+1.
To conclude, we illustrate Propositions 1 and 2 through an example. We assume that
the initial prior over λ is also exponential, so that P0(λ)=1 −e−γλ,F(x|λ)=1 −e−λx and
λ ∈ [0,+∞]. The initial expectation of λ is equal to γ−1. Since λ is the crash rate of the
underlying distributions, γ measures investors’ initial optimism. This stylized example
has the advantage of delivering closed-form solutions. Inserting the expression of p0(λ) in
(2) yields
Pt(λ|kt)=1− e
−λ(kt+γ) (12)
As claimed in Proposition 1, the investors’ beliefs become more skewed towards low
values of λ as output gets bigger. Figure 1 depicts the updating process of Pt(λ|kt) for
given values of the exogenous parameters. The legend in the lower-right corner of Figure
1 reports the time periods attached to each curve. Hence, the investors’ initial beliefs
are depicted by the lowest curve of the graph with a time index equal to 0.T h ec u r v e s
with positive time indexes depict investors’ priors over λ when investment went on for
t periods without experiencing a crash. Figure 1 clearly illustrates the regular shift of
beliefs towards low crash rate distributions.
We can also solve explicitly for the optimal rate of investment. Inserting p0(λ)=
γe−γλ in (9) and solving for the integrals yields
kt+1 − kt =( 2 kt + γ)
µ
β (p + c) − c
c
¶
(13)
Investment is an increasing function of output and therefore of time, as predicted by
Proposition 2. Figure 2 compares the accumulation paths with and without parameter
u n c e r t a i n t y . T h eh o r i z o n t a ll i n er e p r e s e n t st h el i m i ts i z eo ft h em a r k e tz that we have
arbitrarily set to 0.2 for the simulation. The dashed curve depicts the accumulation path
for the set of parameters reported in Figure 1. The plain curve is the accumulation path
that one would obtain in the absence of parameter uncertainty. More precisely, we have
set P0(λ|kt) equal to the Dirac’s delta function for a given λ
∗. To make the two curves
comparable, we have picked λ
∗ equal to the initial expectation of λ, namely γ−1. Then
the investment rates in the ﬁrst period are almost identical in both cases.
After industry capacity has overshot z, ﬁrms stop investing and capacity remains
constant because of the irreversibility constraint. As one might expect, parameter un-
certainty fuels investment as time elapses. More surprising is the rate at which the two
curves diverge. Obviously, the example is stylized and the impact of the learning process
is particularly strong because the lower bound of the crash rate has been set to zero.
60 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
LAMBDA
t=0
t=5
t=10
t=15
Figure 1: Pt(λ|kt) at diﬀerent point in time.
Parameters: γ =0 .1,β =0 .95,c=1 ,p=0 .1.
0 10 20 30 40 50
0
0.05
0.1
0.15
0.2
0.25
TIME
O
U
T
P
U
T
No Parameter Uncertainty
Parameter Uncertainty
Market Limit
Figure 2: Accumulation paths with and without parameter uncertainty.
Parameters: as reported in Figure 1 except for λ
∗ = γ−1 =1 0 .
7Nevertheless, Figure 2 underlines that parameter updating is far from having solely a
second-order eﬀect.
Another issue of interest is the impact of parameter uncertainty on social welfare. As
explained in Rob (1991), competitive ﬁrms ignore the positive informational externality
generated by their investment. Hence, they are too cautious and capacity accumulation
occurs at a slower path than the social optimum. Obviously, parameter uncertainty
increases the informational externality. However, the risk of exceeding the market limit
is also higher, as can be seen from Figure 2. Given that the planner determines the
optimal trade-oﬀ between these two eﬀects, an interesting avenue of research would be to
ascertain whether or not parameter uncertainty worsens the ineﬃciency of the competitive
equilibrium.
4C o n c l u s i o n
The prospects of emerging industries are often quite uncertain. So the prior of investors
over market size is likely to be better described by a mixture than a single distribution.
Parameter uncertainty introduces a new dimension into the learning process. In Rob’s
(1991) model, expectations are that capacity will eventually saturate the market. As the
market grows without reaching this threshold, investors put less and less weight on pes-
simistic predictions. In other words, they become more and more optimistic. According
to this mechanism, when demand is more sustained than what was initially expected,
periods of euphoria similar to the end of the 1990’s naturally arise.
At a technical level, the declining hazard rate assumed in Barbarino and Jovanovic
(2004) can be interpreted as a reduced-form property of the full-ﬂedged investors’ prob-
lem. We have restricted our attention to mixtures of exponential distributions because
only them allow to obtain a conclusion as generic as the one stated in Proposition 2.
Nevertheless, the basic mechanism described in this paper carries over to more elabo-
rated set-ups. Further empirical research would enable us to decide whether the impact
of parameter uncertainty is signiﬁcant, as suggested by the example in section 3.
References
Barbarino, A., and B. Jovanovic (2004) “Shakeouts and Market Crashes”, Unpublished
Manuscript, New-York University.
De Groot, M. (1970) Optimal Statistical Decisions, McGraw-Hill, Inc.
Fudenberg, D., and J. Tirole (1991) Game Theory, Cambridge, Mass.: MIT Press.
Krasker, W.S. (1980) “The ‘Peso Problem’ in Testing the Eﬃciency of Forward Exchange
Markets” Journal of Monetary Economics 6, No.2: 269-276.
Rob, R. (1991) “Learning and Capacity Expansion under Demand Uncertainty” Review
of Economic Studies 58, 655-675.
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