Ambiguity and Dynamic Corporate Investment∗

Lorenzo Garlappi

Sauder School of Business

University of British Columbia

Ron Giammarino

Sauder School of Business

University of British Columbia

Ali Lazrak

Sauder School of Business

University of British Columbia

First version May 2, 2012

This version: February 7, 2013

∗Part of this paper was contained in an earlier working paper titled “Ambiguity, Entrepreneurs andCorporate Finance”. For comments on that paper we thank Murray Carlson, Simone Cerreia Vioglio, BillCooper, Alan Kraus, Fabio Maccheroni, Massimo Marinacci, Jacob Sagi, Tan Wang, and Felipe Zuritafor helpful discussions and seminar participants at Bocconi University, Universidad Catolica de Chile,Copenhagen Business School, the Federal Reserve Bank of Chicago, Queen’s University, The HebrewUniversity of Jerusalem, The Interdisciplinary Centre (Tel Aviv), the University of Aarhus, the Universityof Frankfurt, the University of British Columbia, Tel Aviv University, and Konstanz University for helpfulcomments. We also thank the Social Sciences and Humanities Resaerch Council of Canada for financialsupport.

Ambiguity and Dynamic Corporate Investment

Abstract

We study the effect of ambiguity on corporate financial decisions using a non-Bayesian mul-tiprior approach. We use two models of decision making under ambiguity, Minimum ExpectedUtility (MEU) due to Gilboa and Schmeidler (1989) and Consensus Expected Utility (CEU)due to Bewley (2002), to study a canonical corporate finance model of real investment withexpansion and contraction options. We show that MEU decision makers behave as pessimisticsingle-prior decision makers: they are reluctant to invest and eager to abandon. On the otherhand CEU decision makers act very differently: they too are reluctant to invest but reluctantto abandon projects (escalating commitment). Moreover, anticipation of future reluctance toabandon may induce CEU agents to forego investment in the first place. In this setting, we showhow financial contracts, such as convertible bonds, can be used to make the initial investmentattractive. We also argue that CEU is a reasonable model of group decision making withincorporations and identify related empirical predictions.

1 Introduction

Most of standard finance theory is built on the Bayesian paradigm of subjective expected utility

(SEU) axiomatized by Savage (1954). Despite being the dominant paradigm for the study

of decision making, however, it is well-known that SEU is not equipped to deal with several

phenomena observed in experimental studies. Seminal among these studies are the thought

experiments of Ellsberg (1961) that highlight how ambiguity1 affects individuals’ willingness to

bet.2 Although Ellsberg’s experiments and several other experimental papers emphasize the

subjects’ aversion to ambiguity, there are also studies that have shown situations in which

subjects are ambiguity loving, as in Heath and Tversky (1991) “competence hypothesis” (see

Luce (2000) for an extensive survey of the experimental evidence). Regardless of the direction of

the impact, it is clear that the existing evidence in several fields of study suggests that ambiguity

and attitudes toward ambiguity are important for choice.

In this paper we study how ambiguity affects dynamic corporate decisions. Since corporations

allocate resources across ambiguous and unambiguous investment opportunities, understanding

how ambiguity affects corporate decisions is essential to understanding allocative efficiency.

Moreover, since real decisions directly influence the dynamics of returns, this understanding is

also important for asset prices.

We follow a literature that captures ambiguity through a non-Bayesian multiprior approach

to decision making. There are a number of specific approaches to decision making with mul-

tiple priors that have been developed and from these we select two prominent ones: Minimum

Expected Utility (MEU) due to Gilboa and Schmeidler (1989) and Consensus Expected Util-

ity (CEU) due to Bewley (2002). From these perspectives we reconsider a canonical corporate

finance model of real investment with options to both expand and contract.

The canonical corporate finance model at the heart of our analysis has a very simple struc-

ture. An entrepreneur (E) has a potentially economically valuable idea that involves an initial1Ellsberg refers to ambiguity as “a quality depending on the amount, type, reliability and ‘unanimity’ of

information, and giving rise to one’s ‘degree of confidence’ in an estimate of relative likelihoods.”(Ellsberg, 1961,p. 657).

2One of Ellsberg’s experiments involves two urns with 100 balls each. In the first urn, the unambiguous urn,the subject is told that there are 50 white and 50 blue balls. In the second urn, the ambiguous urn, no informationis given on the proportion of white and blue balls. The subject has to choose an urn and a color. After that,a ball will be drawn from the chosen urn and a prize will be awarded if the drawn ball is of the chosen color.The vast majority of subjects chooses to place either of the bets (blue or white ball) on the unambiguous urn.This behavior cannot be justified by any probability distribution since it implies that the subject believes thatthe probability of a specific outcome (drawing a blue ball from the ambiguous urn) is both less than and greaterthan 50%.

2

investment and a future expansion and contraction option. This problem has been extensively

studied in the Bayesian paradigm of SEU axiomatized by Savage (1954). The fundamental im-

plication of the SEU paradigm is that individuals select among actions with risky outcomes by

attaching a utility index to each outcome and a unique probability to the likelihood that the

outcome will obtain. The decision maker then chooses actions that maximize expected utility

or market value.3

Finance researchers have explored alternative models of rationality or “behavioral models”

although the departure from SEU is not always clear. For the most part, the SEU framework is

maintained but it is assumed that agents do not follow Bayes rule in incorporating evidence or

experience into their decisions.4 We also explore non-Bayesian behavior but do so through an

explicit departure from SEU. Specifically, we follow decision theorists who build their analysis

on the relaxation of specific Savage axioms. We consider two such cases both of which involve

representations of preferences in which a set of probabilities (instead of a unique probability) is

involved.

Perhaps the most widely cited alternative to SEU is MEU, in which the Savage Axiom of

Independence is relaxed. The result is that decisions are characterized by multiple priors and

the decision is determined by the prior that delivers the lowest expected utility. An attractive

feature of MEU preferences is that they can “explain” the decisions observed in the Ellsberg’s

experiments.

An alternative preference framework based on multiple priors is CEU, proposed by Bewley

(2002) as a formalization of Knight’s idea of uncertainty. This preference representation can

be obtained from SEU by first removing the axiom that preferences are complete (i.e., any two

gambles can always be ranked).5 Once the axiom of completeness is dropped the decision maker’s

preferences cannot be represented as being based on a unique prior. Beweley proposes that3Some recent studies have considered how ambiguity aversion affects corporate decisions. In particular,

Nishimura and Ozaki (2007), Riedel (2009),Miao and Wang (2011), and Riis Flor and Hesel (2011) examinethe exercise decision in a real option setting but do so only in an MEU framework. We position our work relativeto this literature below.

4See Baker and Wurgler (2011) for a survey of behavioral corporate finance.5Incomplete preferences were studied originally by Aumann (1962). Dubra, Maccheroni, and Ok (2004) show

that an incomplete order over (unambiguous) lotteries can be represented via a multi-utility representation. Krausand Sagi (2006) extend their analysis to the case of temporal lotteries and provide a model of inter-temporal choicewith incomplete preference that generalize Kreps and Porteus (1978) recursive utility. Ghirardato, Maccheroni,and Marinacci (2004) and Gilboa, Maccheroni, Marinacci, and Schmeidler (2010) provide a general form of theBewley’s representation theorem. Ortoleva (2010) provides a different perspective on Bewley’s inertia assumptionand shows how by starting with a complete set of preferences that exhibit status quo bias one can obtain incompletepreferences as in Bewley.

3

choices be evaluated via a unanimity criterion under which one gamble is preferred to another

if its expected value is higher under all priors within the set of priors. Because these unanimity

preferences are incomplete, there does not exist a numerical index that represents them, making

them unsuitable for optimization problems on which large part of economic decision making is

based. To complete the preferences, Bewley (2002) imposes the assumption of inertia under

which a person remains with the status quo unless an alternative is deemed better under all

priors.6 It is important to note that in Bewley’s characterization, the status quo is not considered

a behavioral bias but a device to complete the preferences. The treatment of the status quo is

conceptually different in the behavioral economics literature that relies on “reference-dependent”

preferences to analyze the implications of biases such as the endowment effect, loss aversion or

framing.7

We feel an examination of CEU is especially appropriate in the corporate context due to its

correspondence to a unanimity rule. That is, CEU can be thought of as a model of a group of

decision makers, perhaps a board of directors or a management team, where each member of

the group has a set of priors. If this group only accepts proposed actions that are acceptable to

all, they will act as if they were a single CEU decision maker.

Our analysis delivers a number of novel interesting findings. We show that MEU and CEU

are observationally equivalent when an entrepreneur faces the decision to expand an existing

venture but have opposite predictions when he is faced with the decision to shut down oper-

ations. Specifically, while MEU implies a “pessimistic” decision rule in both expansion and

shut-down decisions, CEU entrepreneurs are pessimistic in expansions but optimistic in con-

tractions. Because of the unanimity nature of CEU preferences, entrepreneurs expand only if

the scaled up venture is better under the worst possible scenario (similar to MEU). However, in

contractions, for a CEU entrepreneur the worst case scenario is getting rid of a venture that is

very profitable, i.e., CEU takes into account the regret of an action. This implies that, contrary

to MEU entrepreneurs, an CEU entrepreneur may continue operating a project even when they

believe the project may be worth less than the scrap value of the asset under some distributions.6Gilboa, Maccheroni, Marinacci, and Schmeidler (2010) suggest another approach to obtain complete prefer-

ences that relies on imposing axioms for both the unanimous and the MEU preferences and show that MEU canbe seen as a completion of unanimous incomplete preferences.

7See, for example, Kahneman and Tversky (1979), Thaler (1980), Kahneman, Knetsch, and Thaler (1991),and Tversky and Kahneman (1991). Sagi (2006) studies the implications of imposing a “no-regret” axiom on afamily of complete preferences “anchored” at the status quo.

4

This finding is in sharp contrast with with what an MEU model would deliver (see, e.g., Miao

and Wang (2011)).

Perhaps the most interesting and empirically important results of our analysis have to do

with the dynamics of investment decisions. Dynamic models of choice under ambiguity often

result in choices that may not be dynamically consistent. This has been shown to be a problem

in MEU specifications where learning is possible.8 We add to this understanding by showing

that CEU may be time inconsistent due to changes in the status quo over time. Intuitively,

we show that an initial investment in a project creates a status quo that the manager may be

reluctant to leave. Anticipating how the status quo will affect future decisions, the decision

maker may decide not to invest initially, hence avoiding future behavior that is not attractive

at present. We go on to show that under the same set of conditions, a contract with an outside

party, such as a convertible bond, can alter the future status quo in a way that convinces the

manager to undertake the initial investment. Essentially contract design is used to deal with a

“future self”, that is, a future status quo that “current self” would not want to put in place.

Recent studies have applied the multiple prior approach to finance problems. For the most

part, theses applications have been in the asset pricing and portfolio choice areas. Epstein and

Schneider (2010) and Guidolin and Rinaldi (2012) provide an excellent surveys of the literature.

Fewer studies however have considered how ambiguity could affect corporate investment deci-

sions. For example, Nishimura and Ozaki (2007), Riedel (2009), Miao and Wang (2011), and

Riis Flor and Hesel (2011) and examine the exercise decision in a real option setting but do so

only in an MEU framework and do not examine dynamic consistency or contracting.

Applications of CEU in finance include Rigotti (2004) who studies financing decisions, Rigotti

and Shannon (2005) who study risk sharing and allocative efficiency in general equilibrium and

Easley and O’Hara (2010) who use CEU to study liquidity and market ‘freezes’. Our contribution

is to study the effect of different modelling choices of ambiguity (MEU vs. CEU) on dynamic

real investment decisions and their implications for contracting.

In the next section we set out some preliminary elements of our analysis and follow this

with an examination of a two period real option decision. In Section 3 we characterize the

solution of the investment problem and show that different representations of ambiguity lead

to stark difference in observed investment behavior. We also consider the time inconsistency8See, for example, Al-Najjar and Weinstein (2009) and Siniscalchi (2011).

5

in real decisions that can be present in the face of ambiguity. In particular we show that

a decision maker might be willing to invest if they could precommit to a future real option

excercise choice but otherwise would not. In Section 4 we show how a convertible bond can

solve the entrepreneur’s precommittment problem. Section 5 concludes. Appendix A contains

basic results from decision theory that are utilized in our analysis, Appendix B contains proofs

for all propositions, and Appendix C generalizes the contract design analysis.

2 Ambiguity and real investment

We consider a risk-neutral self-financed Entrepreneur (E) who has monopoly access to a project.

The project requires an investment at time t = 0, an expansion or contraction decision at time

t = 1, and delivers cash flows at t = 2. We study the effect of ambiguity on these real investment

decisions by considering E who believes that investments outcomes are governed by a set of prior

distributions. If E the set of priors is a singleton, the outcome is risky but not ambiguous and

E follows Savage’s (1954) SEU criterion to determine his choice. An investment outcome is

ambiguous if the set of priors contains more than one element. In this case we refer to E as

ambiguity sensitive.

In Subsection 2.1, we provide a description of the different types of preferences we consider

in an atemporal context. A more formal review of basic results from decision theory is provided

in Appendix A. In Subsection 2.2 we describe the technology, the model’s information structure,

and the entrepreneur’s actions.

2.1 Preferences

To clearly illustrate the difference between SEU, MEU and CEU preferences, let us consider an

atemporal context where E considers the choice between projects with outcomes in two possible

future states of the world, ‘up’ and ‘down’. In general, E will make decisions based on the belief

that there is a set Π of possible probabilities π characterizing the realization of state ‘up’ (see

6

Theorems 3 and 4 in Appendix A).9 Let us denote the set Π as follows,

Π = {π ∈ [0, 1] : p− ε ≤ π ≤ p+ ε, ε > 0} , (1)

in which ε captures the “degree” of ambiguity. If E adheres to the SEU axioms, then (see

Theorem 2 in Appendix A) he will make his decisions with the belief that there is a unique

(subjective) probability p characterizing the realization of state ‘up’ (i.e. ε = 0). Alternatively,

if E does not adhere to the SEU axioms, then, under both MEU or CEU axioms, he will make

decisions based on the belief that ε > 0.

To analyze the difference among SEU, MEU and CEU we consider a choice between two

“gambles” f and g, i.e., mappings from the state space S = {down, up} to the space of outcomes

Y = R2. In what follows we consider the choice of a risk-neutral decision maker and adopt the

notation Eπ[f ] to indicate the expected value of gamble f under a unique prior distribution

π ∈ Π.

2.1.1 Subjective Expected Utility (SEU)

The preferences of a decision maker who adheres to Savage’s (1954) axioms are represented

via the Subjective Expected Utility (SEU) criterion (see Theorem 2 in Appendix A for a formal

derivation), i.e., given any two gambles f and g,

f � g ⇐⇒ Ep[f ] > Ep[g]. (2)

The above representation makes clear that a SEU decision maker relies on a unique subjective

prior p to choose between alternative gambles.

Figure 1, Panel A, graphically illustrates the SEU criterion for the case of a two-dimensional

state space S. A gamble f is identified by a pair of outcomes (fd, fu) ∈ Y where fd (resp.

fu) is the outcome is state ‘down’ (resp. ‘up’). The downward sloping straight line through f

represents the indifference curve for SEU, i.e., the set of gambles g such that Ep[g] = Ep[f ]. The

dark grey region represents the set of acts g preferred to f , g � f , while the light grey region

represents the set of acts g that are dominated by f , i.e. f � g.9In the robustness literature it is common to refer to the probabilities π as to “models”, i.e., probabilistic

description of a data generating process (see, e.g., Hansen and Sargent (2011)). In this literature, the concept of“prior” is a belief over a model (i.e., a second order belief). Because we do not deal with second order beliefs, inour setting the term “prior” is equivalent to the concept of “model” used in the robustness literature.

7

2.1.2 Minimum Expected Utility (MEU)

The preferences of a decision maker who adheres to Gilboa and Schmeidler’s (1989) axioms, can

be represented via the Minimum Expected Utility (MEU) criterion (see Theorem 3 in Appendix A

for details). Specifically, for this DM, there exists a set of priors Π such that, given any two

gambles f and g,

f � g ⇐⇒ minπ∈Π

Eπ[f ] > minπ∈Π

Eπ[g]. (3)

In the representation (3), the quantities minπ∈ΠEπ[f ] and minπ∈ΠEπ[g] are the utility indexes

for the gambles f and g respectively.

Figure 1, Panel B, provides a diagrammatic representation of MEU preferences. Using the set

of priors (1) and the representation (3), we have that the utility index of the gamble f = (fd, fu)

is

minπ∈[p−ε,p+ε]

Eπ[f ] =

{(p− ε)fu + (1− p+ ε)fd , if fu > fd

(p+ ε)fu + (1− p− ε)fd , if fu < fd(4)

Hence, the MEU indifference curve through f is given by the kinked line in Panel B of Figure 1.

Note that the kink in the indifference curve occurs at a point along the 45-degree line. When

fu > fd the minimum expected utility is determined by the relatively pessimistic prior π − ε

while, when fu < fd the lowest expected utility is determined by the relatively optimistic prior

p+ ε

The dark grey region represent the set of acts g preferred to f , g � f , while the light grey

region represents the set of acts g that are dominated by f , i.e. f � g. From the figure it is

obvious that MEU preferences are complete. For comparison, we also report the indifference

curve through f of a SEU DM (dashed line).

2.1.3 Consensus Expected Utility (CEU)

In an attempt to provide an alternative characterization of Knight’s (1921) distinction between

risk and uncertainty, Bewley (2002) develops a theory of choice under uncertainty (ambiguity)

that starts from Savage’s (1954) SEU axioms and drops the axiom of completeness. Once

this axiom is removed, the decision maker’s preferences cannot be represented via a unique

probability distribution. Instead, the choice between two gambles can be represented via a

unanimity criterion (see Theorem 4 in Appendix A for details). Under this “unanimity” criterion,

8

a gamble f is preferred to a gamble g if its expected value is higher under all probability

distributions in Π, i.e.,

f � g ⇐⇒ Eπ[f ] > Eπ[g], ∀π ∈ Π, (5)

or, equivalently,

f � g ⇐⇒ minπ∈Π

Eπ [f − g] > 0. (6)

Figure 1, Panel C provides a diagrammatic representation of unanimity preferences in the case

of a two-dimensional state space S. The dark grey region represents the set of gambles g that

are preferred to f . According to (5), this region is the intersection of the “better-than-f” sets

of an SEU DM with prior p− ε and of an SEU DM with prior p+ ε. A similar observation can

be made for gambles that are worse than f , represented by the light grey region in the figure.

For comparison purposes we report also the indifference curve of the SEU DM (dashed line).

Comparing Panels B and C, it is immediate to see the incomplete nature of the unanimity

ordering. The un-shaded regions in Panel C contain gambles g that are not comparable to f :

there are priors π′ 6= π′′ for which Eπ′ [g] > Eπ′ [f ] and Eπ′′ [g] < Eπ′′ [f ]. Unlike SEU and MEU,

the unanimity criterion does not assign a unique “value” to a gamble and therefore cannot always

specify what the decision maker will do when faced with a choice. To resolve this indeterminacy

for binary choices Bewley introduces the following inertia assumption:

Inertia assumption. A Decision Maker identifies one of the alternatives as the status quo

and will only accept an alternative gamble if the expected value of the alternative is strictly better

than that of the status quo under all possible priors.

In the rest of this paper we will refer to Consensus Expected Utility or CEU as the unanimity

criterion (5) augmented by the Inertia assumption. Assuming, for example, that f is the status

quo in Panel C of Figure 1, and that g is a gamble in the interior of one of the un-shaded

regions. Then the acts f and g are not comparable with the unanimity rule. Due to the inertia

assumption, the CEU DM will then choose to remain with the status quo f rather than moving

to g.

9

Figure 1: SEU, MEU and CEU preferences

The figure reports the indifference curve through the gamble f for SEU (Panel A) and MEU (Panel B).Panel C reports the preference ordering according to CEU. In all panels, the dark grey region indicates“better-than-f” gambles and the light grey region indicates “worse-than-f” gambles.

Panel A: SEU

up

45°!"p

!"p

!f

g " f

g " fg # f

g # f

down

Panel B: MEU

up

45°

!"p#$

!"p%$

!f

g " f

g " fg # f

g # f

down

Panel C: CEU

up

45°

!"p#$

!"p%$

!f

g " f

g " f

g # f

g # f

down

10

Notice that the CEU cannot always predict the behavior when facing non binary one shot

decisions. Assume, for example, that f is the status quo in Panel C of Figure 1, and that the DM

is contemplating the choice of the gambles g and h in the shaded upper cone of “better-than-f”

gambles. Then both choices of f and g are preferred to the status quo f and as a result, both

choices are consistent with CEU. The potential for indeterminate choices under CEU preferences

was coined by Bewley (2002) as “indeterminateness”.10

2.1.4 CEU as a model of group behavior

We believe that CEU is an appropriate description of group behavior where the group contains

individuals who may hold unique or multiple priors over the outcomes of an action. Consider

a board of directors evaluating a project that is seen as an alternative to what they jointly

consider to be the status quo. Suppose that each individual in the group held a set of beliefs

Πi and define ΠG =⋃iΠ

i as the union of all sets of priors of all individuals i in the group. If

the group will only accept alternatives if they are unanimously agreed to, they will behave as

if they were the single agent CEU decision maker described above who holds the set of priors

ΠG. This is because the alternative action will be evaluated relative to the status quo for all

elements of the group’s set of priors.

This interpretation of CEU suggests many empirical questions that could be related to

investment and financing decisions. For instance:

1. Where do the multiple priors come from? Studies of corporate decisions such as investment

and financing policy indicate that these decisions depend on personal characteristics such

as age, education, and whether or not the decision maker was a “depression baby”.11 In

this literature age and having lived through the depression are seen as being related to

conservative preferences or risk aversion in an SEU setting. Our alternative interpretation

is that these individuals see multiple priors when they evaluate decisions, having seen a

world that younger colleagues might feel is impossible.10The indeterminateness can sometime be eliminated by allowing the DM to contemplate future rounds of

choices moving from one undominated gamble to another. For example, if we allow one more round of choicesby giving the DM the power to move from f to g, and then given the new status quo of g contemplate a moveto h. If h dominates g in the second round then under CEU preferences, the DM should end up with the gambleh. However, if g is not comparable to h the CEU DM can end up either with g or with h and as a result, bothchoices g and h are consistent with CEU, giving rise to a choice indeterminateness.

11See for example Bertrand and Schoar (2003), and Malmendier and Tate (2005).

11

2. How does a group define the status quo? For instance, when faced with the choice of

paying a fixed cost to continue producing relative to shutting down a factory, the group is

not able to define the status quo as “doing nothing”. Hence, the definition of the status

quo is itself a group decision.

3. Do groups such as boards follow a unanimity rule? Although boards may formally govern

by majority rule, do they, as an empirical matter, only make decisions when there is

unanimous agreement?

2.2 Technology and information structure

We carry out the analysis in the context of a simple two-period information structure. The

state of the world evolves over the two periods according to a binomial process. Starting at the

unambiguous value s0, the process evolves to s̃1 at time t = 1 where the random state s̃1 takes

the value su (‘up state’) or sd (‘down state’). Conditional on being in the ‘up state’ (resp. ‘down

state’) at time t = 1, the state in the second period is denoted by s̃2 = s̃u2 (resp. s̃2 = s̃d2) where

s̃u2 (resp. s̃d2) takes the value suu (resp. sdu) in the ‘up-up state’ (resp. ‘down-up state’) or sdu

(resp. sdd) in the ‘up-down state’(resp. ‘down-down state’). We assume that

su > sd, and suu > sud > sdu > ddd. (7)

To simplify exposition, we assume that the same set Π describes conditional one-step-ahead

priors at time t = 0, π0, and at time t = 1 in both ‘up’ and ‘down’ states, πu and πd, respectively.

This assumption essentially imposes independence between successive realizations of the state.12

This in turns implies that, upon observing the realization of s̃1, E can only refine the set of

future paths (binomial subtrees) but cannot learn more about the law of motion governing the

random variable s̃2.13

We study decisions made by E at two different points in time. The first decision is whether

or not to invest I0 to acquire one unit of capital at t = 0. If the initial investment is made, E12A concrete way to understand our assumptions on ambiguity is to assume that nature flips a first coin at time

t = 1 to determine the current shock (up or down). At time t = 2 a second independent coin is used to determineif the second shock is an up or a down movement. Both coins are seen as ambiguous with a set of priors given in(1).

13Because we take the one-step-ahead priors as primitive in our description of ambiguity, the resulting uncon-ditional set of priors over the state space satisfies by construction the rectangularity condition of Epstein andSchneider (2003) which guarantees the dynamic consistency of MEU preferences.

12

faces a second decision at t = 1, when the scale of the initial investment can be maintained,

expanded or contracted. Let A = {E , C,R} be the set of possible actions at time time t = 1,

where E indicates Expansion, C indicates Continuation, and R indicates Contraction. If the

initial investment is made at t = 0 the project delivers cash flows C̃2 at time t = 2 that depend

on the state of nature, s̃2, and the scale of the firm, λ. Specifically,

1. E requires payment of the amount I1 at t = 1 to increase the scale of the firm to λ > 1

and generates the cash flow C̃2 = λs̃2 − I1 at time t = 2.

2. C requires a payment m where m can be thought of as maintenance or the second round of

investment required to put the capacity in place and, generates the cash flow C̃2 = s̃2−m

at time t = 2.

3. R involves scrapping the firm by setting λ = 0 and receiving the recovery value of C̃2 = R

at time t = 2.

Figure 2 provides a diagrammatic representation of the choice available at time t = 1 if

s̃1 = sj , j = u, d. The point f denotes the cash flow sju, sjd, obtained at t = 2 if the entrepreneur

could continue operations at the scale λ = 1. We assume, however, that continuation requires a

payment for maintenance/completion, m, resulting in a continuation value of f −m. Because

of (7), f lies above the 45-degree line.

A convenient way to understand a change in scale (expansion or contraction) is to consider

the ray going through f (dashed line in the figure). The decision to expand at time t = 1

corresponds of moving upward along the ray from f to λf , λ > 1, while the decision to contract

corresponds to moving to point R on the 45-degree line where the payoffs are state independent.

If the entrepreneur decides to expand, it will have to pay an unambiguous cost I1 to obtain a

“scaled” up version of the current firm. The expansion decision therefore entails comparing the

cash flow from current operations f , net of maintenance costs m, with the alternative λf − I1.

By inequality (7) and the fact that the expansion cost I1 is unambiguous, the “gamble” λf − I1

lies always to the left of the ray going through f .14 If the entrepreneur decides to contract, he

selects a scale λ = 0 and receives an unambiguous recovery value of R. Hence, the contraction

decision entails comparing the cash flow from current operations f net of maintenance costs14The opposite is true, obviously, if inequality (7) were to be reversed.

13

Figure 2: Expansion and contraction decisionsThe figure illustrates the expansion and contraction decisions at time t = 1, where f is generic notationfor s̃j

2, j = u, d.

ju

45°

!f!f!m

!"f, "#1

!"f!I1!Expansion"

!

!R!Contraction"

jd

m with the alternative R on the 45-degree line. By inequality (7) cash flows corresponding to

contraction decisions will always lie to the right of the ray going through f .15

3 Solutions of the dynamic real investment problem

We solve the investment decision problem recursively determining first the state dependent

expansion, continue or contraction decision at time t = 1 and then the initial investment decision

at time t = 0. In our analysis we focus on the difference between the solutions obtained under

SEU, MEU and CEU.

3.1 Time 1 decisions

Figure 3, Panel A, describes the expansion, continue, and contraction choice under MEU prefer-

ences at time t = 1 and compares it to SEU. The figure combines the preference description in

Figure 1, with the technology description in Figure 2. The point f −m represents the action C,15In general we could also consider a partial contraction, in which E selects a scale λ ∈ [0, 1) and receives a

scrap value R in exchange of selling a fraction 1−λ of the firm. Graphically this would entail comparing the cashflow from current operations f −m with the alternative λf + R, λ ∈ [0, 1). Because the contraction payoff R isunambiguous, the gamble λf +R lies always to the right of the ray going through f .

14

Continue, which provides the cash flow of one unit of capital at time t = 2 in each state ‘up’ or

‘down’. For convenience we use the generic notation f for the cash flow s̃u2 and s̃d2. The kinked

line is the MEU indifference curve through f −m, while the dotted line is the indifference curve

for a SEU DM.

The SEU decision is simply

Expand �SEU Continue ⇐⇒ λEp[f ]− I1 > Ep[f ]−m (8)

Contract �SEU Continue ⇐⇒ R > Ep[f ]−m, (9)

where the E makes use of the unique probability p in assessing alternatives.

From the figure we clearly see that if E has MEU preferences she will use the prior π = p− ε

to decide whether to expand or contract. There is a sufficiently high level of investment cost

I1 and a sufficiently low level of maintenance cost m for which and MEU DM will not expand

while an SEU DM will. Similarly, there are scrap values R for which MEU will contract while

an SEU will not. The choice of a MEU DM are then

Expand �MEU Continue ⇐⇒ λEp−ε[f ]− I1 > Ep−ε[f ]−m (10)

Contract �MEU Continue ⇐⇒ R > Ep−ε[f ]−m, (11)

In summary, the analysis in Figure 3, Panel A, shows that for the expansion and contraction

decisions, if E has MEU preferences she behaves as if she were a pessimistic SEU DM who has

beliefs captured by the prior p− ε.

Panel B of Figure 3 illustrates the key difference between MEU and CEU preferences. CEU

requires that the status quo be specified and we assume the action C (Continue, i.e. the point

f−m) is considered to be the status quo. As in the MEU case, the relevant prior that determines

whether an expansion decision is taken or not is π = p − ε. However, the relevant prior for

deciding whether to contract or continue is now π = p + ε and no longer π = p − ε, as it is for

the case of MEU preferences. When offered the option to sell an ambiguous firm, the CEU DM

considers the best possible scenario as a criterion to decide whether to surrender the firm for an

15

Figure 3: Time 1 decisions

Panel A (B) reports the expansion/contraction decision under MEU (CEU) preferences. In both panels,the dotted line refers to the indifference curve under SEU preferences. f is generic notation for s̃j

2,j = u, d.

Panel A: MEU

up

45°

!"p#$

!"p%$SEU

SEU

!f!

f#m

!&f, &'1

!&f#I1!Expansion"

!

R!Contraction"down

Panel B: CEU

up

45°

!"p#$

!"p%$SEU

SEU

!f!

f#m

!&f, &'1

!&f#I1!Expansion"

!

R!Contraction"down

16

unambiguous amount of cash R. The choice of a CEU DM are then

Expand �CEU Continue ⇐⇒ λEp−ε[f ]− I1 > Ep−ε[f ]−m (12)

Contract �CEU Continue ⇐⇒ R > Ep+ε[f ]−m, (13)

In summary, the analysis in Figure 3 shows that although it is difficult to distinguish the

expansion decision of an MEU and CEU decision maker from a pessimistic SEU DM, the con-

traction decision delivers very different results. While the MEU contraction decision (11) is

indistinguishable from SEU contraction (9) with a unique “pessimistic” prior π = p − ε, the

CEU decision is clearly different from both SEU and MEU. A comparison of (11) with (13)

shows that when facing a contraction an MEU decision maker is pessimistic while a CEU deci-

sion maker is optimistic. Hence, even when E can sell a firm for R and may even consider R to

be larger than his expected payoff from continuing under some priors, he will not contract if he

entertains just one prior under which the expected continuation value of the firm is higher than

R.

It has been observed that managers are reluctant to divest or shut down projects that have

not done well. These observations have motivated explanations based on agency problems,

reputation concerns and asymmetric information (see, for example, Boot (1992) and Weisbach

(1995)). CEU can provide an alternative explanation in terms of ambiguity even in the absence

of asymmetric information. If DMs are ambiguity sensitive and adhere to the CEU axioms they

will be reluctant to terminate a project because of the importance that potential regret plays in

their decision making.

A related literature has shown that groups and not just individuals seem to continue projects

that some members of the group felt should be terminated. For example Guler (2007) studies

venture capital syndicates and finds that the probabililty of success and the eventual returns to

investments decline with the number of financing rounds and that the probability of terminating

an investment at any stage decreases as the number of members of the syndicate increases. This

is explained as an example of political and institutional influences on investment decisions. The

alternative that CEU provides is that as the number of syndicate members increases, ΠG, the

groups set of priors, increases and as a result, the effective amount of ambiguity, ε also increases.

With more priors to satisfy, leaving the status quo becomes less likely.

17

3.2 Time 0 decision

At time t = 0 E is endowed with an unambiguous quantity I0 and faces the choice of whether

to invest or not. Let us define an investment plan as a pair ℘ = (au, ad) ∈ P where P = A×A

is the set of all possible investment plans and where for each j ∈ {‘up’, ‘down’}, aj is the action

in state j at time 1. For instance, we indicate by (E , E) the investment plan: ‘Expand in the

up state, Expand in the down state’, and similarly for all the other 8 possible investment plans.

We denote by ℘0 the plan in which E does not invest at time t = 0.

The outcome of every investment plan ℘ = (au, ad) is a random time t = 2 payoff

C̃℘ =

{Cau(s̃u2) when s̃1 = su

Cad(s̃d2) when s̃1 = sd, (14)

where Caj (·), j = u, d, are deterministic functions defined by

Caj (x) =

λx− I1 if aj = Ex if aj = CR if aj = R

. (15)

Notice that C̃℘ is the unconditional payoff of an investment plan, i.e., a random variable with

four possible outcomes, while Cau(s̃u2) and Cad(s̃d2) are payoffs conditional on the state realized

at time t = 1, i.e., they are random variables with two possible outcomes.

The choices expressed by E at time t = 1, characterized in Section 3.1, restrict the set of

possible investment plan that E will consider at time t = 0. We refer to an implementable

investment plan as one that E will decide to carry out at time t = 1. We now formally define

this set for each of the preferences considered above.

Definition 1. An investment plan ℘ = (au, ad) ∈ P is

1. SEU-implementable if

Ep[Caj (s̃j2)] ≥ Ep[Ca′j (s̃j2)], ∀a′j ∈ A, j = u, d. (16)

We denote by P∗SEU the set of all SEU-implementable plans.

18

2. MEU-implementable if

minπ∈Π

Eπ[Caj (s̃j2)] ≥ minπ∈Π

Eπ[Ca′j (s̃j2)], ∀a′j ∈ A, j = u, d. (17)

We denote by P∗MEU the set of all MEU-implementable plans.

3. CEU-implementable if

Eπ[Caj (s̃j2)] > Eπ[CC(s̃j2)] = Eπ[s̃j2], ∀π ∈ Π, and ∀aj 6= C, j = u, d. (18)

We denote by P∗CEU the set of all CEU-implementable plans.

Note that the definition of CEU-implementability depends on the DM’s status quo at time

t = 1. In our real investment problem, the status quo is the action C at time t = 1 in each state

j = u, d and therefore, as stated in (18), a plan is implementable if its conditional payoff is in

the better-off set with respect to the status quo C. Moreover, the status quo plan (C, C) is by

construction CEU-implementable because condition (18) is not restrictive when (au, ad) = (C, C).

For SEU and MEU, the notion of implementability does not depend on the status quo and

conditions (16) and (17) require global optimization rather than comparability with respect to

the status quo.

Given an implementable plan ℘ = (au, ad), the decision of whether to invest or not at time

t = 0 is similar to that described at time t = 1. Specifically, given any implementable investment

plan the SEU DM will choose the investment plan ℘ at time t = 0 if

Ep[C̃℘] > I0, (19)

where Ep[C̃℘] = p2Cau(suu) + p(1− p)Cau(sud) + (1− p)pCad(sdu) + (1− p)2Cad(sdd).

An MEU DM with priors π0 ∈ Π, πu ∈ Π and πd ∈ Π, with the prior set Π as described in

Section 2.2, will choose the investment plan ℘ at time t = 0 if

min(π0,πu,πd)∈Π3

Eπ0,πu,πd[C̃℘] > I0, (20)

where

Eπ0,πu,πd[C̃℘] = π0[πuCau(suu)+(1−πu)Cau(sud)]+(1−π0)[πdCad(sdu)+(1−πd)Cad(sdd)] (21)

19

A CEU DM will choose the investment plan ℘ at time t = 0 if

Eπ0,πu,πd[C̃℘] > I0, for all (π0, πu, πd) ∈ Π3. (22)

If there is at least an implementable plan ℘ that is chosen at time t = 0, then E will invest

at time zero. We refer to a plan thus chosen as to a recursive solution of the dynamic real

investment problem. If there are more than one plan ℘ that E will choose, then a potential

indeterminateness can arise. For the case of complete preferences such as SEU and MEU such

indeterminateness can be resolved by choosing the plan with the highest value (i.e., the plan(s)

that maximizes expected payoff for SEU or the plan(s) that maximize the minimum expected

payoff for MEU). For the case of CEU preferences we may be left with an indeterminateness as

we discussed in Section 2.1.3.

We illustrate the construction of the set of implementable investment plans and the solution

of the dynamic investment problem with the help of Figure 4. For simplicity we henceforth

assume m = 0. In the figure we assume that the recovery value R and expansion cost I1 satisfy

the following conditions:

Assumption 1. The recovery value R for shutting down the firm at t = 1 is such that

Ep−ε[s̃d2] < Ep[s̃d2] < R < Ep+ε[s̃d2]. (23)

The investment cost I1 for expanding the scale to λ at time t = 1 is such that

λEp[s̃d2]− I1 < Ep[s̃d2], and Ep−ε[s̃u2 ] < λEp−ε[s̃u2 ]− I1 (24)

Condition (23) implies that, in the ‘down’ state, R is undertaken by SEU and MEU but not

by CEU who prefers the status quo C (see Panel B of Figure 4). The first inequality of (24) implies

that in the ‘down’ state E is not undertaken under any preference. The second inequality in (24)

implies that, in the ‘up’ state, E is undertaken under all preferences (see Panel A of Figure 4).

In this example, the set of SEU- and MEU-implementable plans coincide, i.e., P∗SEU = P∗MEU =

{(E ,R)} while the set of CEU implementable plans is P∗CEU = {(E , C), (C, C)}.

20

Figure 4: Time 1 decisions: MEU vs CEUThe shaded area in each panel represents the set of gambles over which MEU and CEU will disagree.The dotted line represents the SEU indifference curve through s̃u

2 (Panel A) and s̃d2 (Panel B).

Panel A: time t = 1, ‘up’ state

up

45°

!"p#$, MEU"CEU

!"p%$, CEU

!"p%$, MEU

SEU

SEU

!s&2u

! 's&2u, '(1

!'s&2

u#I1!Expansion"

!R!Contraction"

down

Panel B: time t = 1, ‘down’ state

up

45°

!"p#$, MEU"CEU

!"p%$, CEU

!"p%$, MEU

SEU

SEU

!s&2d

! 's&2d, '(1

!'s&2

d#I1!Expansion" !

R!Contraction"down

21

Because the implementable plan under SEU and MEU is unique, applying the choice crite-

rion (19) we have that a SEU DM invests at time t = 0 if I0 < Ep[C̃(E,R)], i.e., if

I0 < p(λEp[s̃u2 ]− I1) + (1− p)R. (25)

By the criterion (20) a MEU DM invests at time t = 0 if I0 < min(π0,πu,πd)∈Π3 Eπ0,πu,πd[C̃(E,R)].

Under the condition of Assumption 1 this implies

I0 < (p− ε)(λEp−ε[s̃u2 ]− I1) + (1− p+ ε)R. (26)

By the criterion (22) a CEU DM invests at time t = 0 if I0 < Eπ0,πu,πd[C̃℘] for all (π0, πu, πd) ∈ Π3

for either ℘ = (E , C) or ℘ = (C, C). Under the condition of Assumption 1 this implies that

investment occurs if either

I0 < (p− ε)(λEp−ε[s̃u2 ]− I1) + (1− p+ ε)Ep−ε[s̃d2], (27)

or

I0 < (p− ε)Ep−ε[s̃u2 ] + (1− p+ ε)Ep−ε[s̃d2]. (28)

If both conditions (27) and (28) are satisfied, there is indeterminateness at time t = 0. Com-

paring the MEU investment condition (26) to the CEU investment condition (27) we see that,

because Ep−ε[s̃d2] < R by Assumption 1, for values of the investment cost I0 in the interval

(p− ε)(λEp−ε[s̃u2 ]− I1) + (1− p+ ε)Ep−ε[s̃d2] < I0 < (p− ε)(λEp−ε[s̃u2 ]− I1) + (1− p+ ε)R (29)

an SEU and MEU DM would invest while a CEU DM would not. This happens because the

above recursive solutions restrict the choice of plans to only those that are implementable at

time t = 1. For a CEU DM this restriction excludes investment plans that, if implementable

would be attractive. In fact, if we consider the plan (E ,R) and apply the choice criterion (22)

we can see that, under condition (29) this plan will be undertaken by the CEU entrepreneur if

it were implementable because I0 < Eπ0,πu,πd[C̃(E,R)] for all (π0, πu, πd) ∈ Π3. However because

(E ,R) 6∈ P∗CEU the DM will not consider such a plan when constructing its investment problem

recursively. In other words, if the CEU DM were able to precommit to the plan (E ,R) he would

undertake it but such plan is not an implementable plan. For the CEU DM the implementability

22

restriction acts as a constraint that prevents him from undertaking actions at time t = 1 that he

would consider attractive at time t = 0. This happens because CEU preferences are expressed

with respect to a status quo and raises the important issue of time consistency to which we now

turn.

3.3 Dynamic consistency

The above example highlighted an important tension between the “time-0” and “time-1” en-

trepreneur with CEU preferences. The time-0 entrepreneur understands that the plan (E ,R)

implies that his status quo will change from ℘0 at time t = 0 to (C, C) at time t = 1. Under the

conditions of Assumption 1, the time-0 entrepreneur then knows that the time-1 entrepreneur

will no longer consider the choice R in the ‘down’ node attractive, thus making the plan (C,R)

not implementable. On the other hand, if the time-0 entrepreneur were able to precommit to

the plan (C,R) he will chose to undertake it. The example illustrates that, the passage of time

and the resulting change in status quo, induces a preference reversal for the CEU DM. This

reversal is not happening with SEU or MEU preferences under the rectangularity condition on

the prior imposed in Section 2.2.

In this subsection we formalize the concept of dynamic consistency. In the next section we

illustrate how the time-0 entrepreneur can issue securities such as convertible bonds to relax the

implementability constraint and therefore remove the time inconsistency issue.

Definition 2. Let Q = P∪{℘0} be the set of all possible investment plans P augmented with the

‘Do not invest’ plan ℘0, and let P∗ be the set of implementable plans with respect to either SEU,

MEU or CEU preferences, as in Definition 1. A preference ordering is dynamically consistent

if for every plan q ∈ Q the following condition is satisfied:

If q � ℘ ∀℘ ∈ P∗ with ℘ 6= q then 6 ∃℘′ ∈ P∗ for which ℘′ � q, (30)

where P∗ is the set of investment plans that are not implementable.

In other words, if there are no implementable plans that are preferred to the plan q, then

dynamic consistency requires that there cannot be non-implementable plans that are preferred to

q. The following proposition shows that SEU and MEU preferences are dynamically consistent.16

16It is well known (see, for example, Al-Najjar and Weinstein (2009) and Siniscalchi (2011)) that, under a generalinformation structure, MEU preferences can lead to time inconsistent choices. As we discuss in footnote 13, our

23

Proposition 1. Both SEU and MEU preferences satisfy the dynamic consistency condition (30).

As we show in Appendix B dynamic consistency of SEU and MEU follows because any non

implementable plan is dominated by a maximal implementable plan.

Unlike SEU and MEU, CEU preferences are not dynamically consistent. The example of the

previous section illustrates that condition (30) is violated for CEU preferences because the plan

(E ,R) is non-implementable but preferred to not investing (i.e., the plan ℘0) at time 0 while

the only implementable plan, (E , C) is not preferred to ℘0.

To understand the source of dynamic inconsistency for CEU preferences let us consider a

dynamic choice problem in which the DM at time t = 0 has to choose between gambles f0 and

f1 and, at time t = 1, he has to choose between gambles f1 and f2. All gambles are defined on a

two dimensional state space. To fix ideas we can think of a simplified version of the investment

problem discussed in the previous section in which we remove the ‘up’ node in the decision tree.

Without loss of generality let us assume that f0, the status quo at time t = 0, is an unambiguous

gamble. Figure 5 reports f0 on the 45-degree line. The solid line with a kink at f0 represents

the contour of the set containing gambles that are preferred to f0. In the figure we also report

the gamble f1, which represents the status quo at time t = 1.

The shaded area in the figures denotes the regions of gambles with t = 2 payoffs that will

give rise to dynamically inconsistent choices. To see this, consider first the recursive solution.

To obtain this solution let us start from the decision at time t = 1 and compare the status quo

f1 to f2. Because f2 6� f1, the DM at time t = 1 will stick with the status quo f1, which is the

only implementable plan. Given this, the DM at time t = 0 will then have to compare f0 with

f1. Because f1 6� f0 the time-0 DM will stick with the status quo f0. Hence f0 is the recursive

solution for the dynamic problem portrayed in Figure 5.

Let us now suppose that at time t = 0 the DM can precommit to either gamble f1 or

f2. Because f1 6� f0, f1 will not be chosen over f0. However f2 � f0, and it will be the

precommitment solution of the dynamic choice problem. The pre-commitment solution f2 and

the recursive solution f0 disagree. In general, therefore, as stated in Definition 2, dynamic

inconsistency arises when: (i) f1 6� f0 and (ii) f2 6� f1, and (iii) f2 � f0. Note that these

conditions imply a violation of the dynamic consistency condition (30). Intuitively, dynamic

information structure is built by taking the one-step-ahead priors as primitive in our description of ambiguity. Asshown by Epstein and Schneider (2003), the resulting unconditional set of priors over the state space will satisfiesthe “rectangularity condition” that guarantees time consistency of MEU preferences.

24

Figure 5: Dynamic inconsistency of CEU preferencesThe highlighted region in the figure refers to acts at time 2 that will lead to disagreement betweencommitment and recursive solution.

up

45°

!f0

!f1

!f2

down

inconsistency arises when there is a change in status quo over time and the two status quibus

cannot be ranked. The shaded region in Figure 5 illustrates the set of gambles f2 for which

dynamic inconsistency arises. From the figure it is also clear to see that such inconsistency

will not arise with SEU preferences because the indifference curves cannot cross. For MEU

preferences dynamic inconsistency does not arise provided that there is no updating in the

priors. If prior changes over time due to learning, it is however possible that MEU indifference

curve cross, giving rise to the dynamic inconsistency similar to that detected in the case of CEU

preferences.

Finally, figure 5 also helps us understand how contracts can help resolve the dynamic incon-

sistency problem faced by a CEU entrepreneur. By signing contracts at time 0, the time-0 DM

can affect the payoff of the gamble f2. A contract can resolve inconsistency if it can manipulate

either the status quo f1 and/or the payoff of the gamble f2, so that dynamic consistency can

be achieved. In the next section we show how derivative contracts can be useful tools for the

“time-0” entrepreneur to discipline the decisions of his “time-1” self. Because SEU and MEU

preferences are time-consistent under our information structure, we focus the rest of the section

on CEU preferences.

25

4 Convertible bonds as a precommitment mechanism

An implication of the time inconsistency discussed in Section 3.2 is that there is an investment

plan that the entrepreneur would like to undertake at time t = 0 but which is not implementable

at time t = 1. As a result, if the entrepreneur were able to precommit to the non-implementable

plan, he would undertake the investment at t = 0.

The specific example we used was one where (E , C) was implementable but not acceptable at

time t = 0 while (E ,R) was non-implementable but acceptable at time t = 0. One possible way

to precommit to (E ,R) and therefore making it both acceptable and implementable is to trade

a security that will provide the incentives to shut down the firm if the ‘down’ state occurs at

time t = 1. In the sequel, we show that securities with convertible features, such as convertible

bonds, can be used to achieve this objective, thus making the investment decision attractive as

of time zero.17

Assume the CEU entrepreneur who faces the technology restriction in condition (29) issues

at time t = 0 a convertible bond with face value X and a conversion ratio 0 ≤ α ≤ 1. The terms

of the contracts are such that the bondholder can decide to convert the bond into α units of

the firm at time t = 2, after the entrepreneur has decided whether to continue, expand or shut

down. The bondholder payoff at maturity is

max{αC̃2, B̃2} (31)

where B̃2 = min{X, C̃2} is the payoff of straight bond and where C̃2 = R if the project is shut

down at time t = 1, C̃2 = s̃i2, i = u, d if the project is kept at the initial scale at time t = 1 and

C̃2 = λs̃i2 − I1, i = u, d, if the entrepreneur decides to increase the scale of operation at time

t = 1 in state i = u, d. Let assume that, in addition to Assumption 1, the investment cost I1,

and the expansion scale λ satisfy the following condition

sdu <I1

λ− 1< sud. (32)

17In Appendix C, we show that the issue of an action dependent security—i.e., a security that pays off only ifthe firm is shut down or one that pays off ony if the firm is operating—can more directly provide the incentivesto to shut down the firm in bad time. While this type of action dependent contracts are not observed in practicethey help understand why a security like a convertible also provide the desired incentives.

26

As we will see in the proof of Proposition 2, this condition is sufficient, although not necessary,

to guarantee that the incentive to expand the scale of the firm are not distorted by the presence

of the convertible security.

The following proposition characterizes the set of convertible bonds that the entrepreneur

can issue to insure that the investment plan that is preferred at time at time t = 0 is carried

through at time t = 1.

Proposition 2. Suppose that Assumption 1 and conditions (29) and (32) are satisfied. Then

by issuing a convertible bond with conversion ratio α ∈ (0, 1) and face value X such that

α sdd + max{

0,(1− α)Ep+ε[s̃d2]−R

1− p− ε

}< X < min{sdd, αR}, (33)

the entrepreneur will invest at time 0, expand the firm in the ‘up’ state and shut down in the

‘down’ state. As a result the presence of the contract will induce the entrepreneur to undertake

the investment at time 0, contrary to his preferred choice when no convertible bond is issued.

The intuition for the proposition is as follows. Under the stated assumption, the optimal

conversion policy of the bondholders “penalizes” the entrepreneur for continuing operations in

the ‘down’ state thus providing the incentive to shut down the firm. Specifically, under condition

(33), if the entrepreneur continues in ‘down’ state, the bond will be converted in ‘down-up’ but

not in ‘down-down’ and the entrepreneur’s residual payoff at time t = 2 will be

ξ̃2 =

{(1− α)sdu in state ‘down-up’sdd −X in state ‘down-down’

(34)

If the entrepreneur shuts down in the ‘down’ state, his residual payoff at time t = 2 will be

(1−α)R. In state ‘down’ when deciding whether to continue or shut down, the entrepreneur will

compare the residual payoff ξ̃2 with (1−α)R. The payoff ξ̃2 can be rewritten as ξ̃2 = (1−α)s̃d2−ζ̃,

where the random variable

ζ̃ =

{0 in state ‘down-up’X − αsdd > 0 in state ‘down-down’

(35)

27

can be interpreted as a penalty induced by the bondholders conversion decision. When the

penalty ζ̃ = 0, the entrepreneur must compare the residual payoff (1 − α)s̃d2 with the residual

payoff (1−α)R. Under Assumption 1, Ep−ε[s̃d2] < R < Ep+ε[s̃d2] and hence the CEU entrepreneur

will not shut down the firm. For sufficiently large penalty ζ̃, the entrepreneur may reverse his

choice and shut down the firm. The left hand side of condition (33) quantify the magnitude of

the penalty that is necessary to reverse the entrepreneur’s choice.

The convertible bond considered in Proposition 2 is not the unique way to solve the time

inconsistency problem of a CEU entrepreneur. In general it is in fact possible to engineer

multiple securities with the right combination of call and convertible features in order to provide

the desired incentives in the future states of the world. Our emphasis on the convertible bond

is motivated by the empirical fact hat such contract are frequently observed in practice during

early stage financing of start-ups and R&D ventures. These are environments in which concerns

about ambiguity are of first order importance. In the same spirit as Mukerji (1998), who claims

that ambiguity aversion can explain the existence of incomplete contracts, Proposition 2 suggests

therefore that entrepreneurs’ incomplete preferences can explain the use of convertible bonds in

the early phases of life of ventures where ambiguity about future prospect is most severe.

Finally, it is important to stress that the dynamic real investment problem and the contrac-

tual solution to time inconsistency, clearly illustrate the conceptual difference between choice

induced by CEU preference and the role of transaction costs. While in a static choice problem,

CEU choices are observationally equivalent to those expressed by a SEU agent who faces trans-

action costs (i.e., bid-ask spreads), in a dynamic context this analogy necessarily breaks down.

This happens because SEU preferences are dynamically consistent, even in the presence of trans-

action costs, while CEU preferences can be dynamically inconsistent because of the change of

status quo. Therefore, the contractual solution of Proposition 2 will not reverse the preferences

of a SEU entrepreneur who does not invest because of the high transaction costs he may face in

shutting down the firm in the future.

5 Conclusion

In this paper we examine a simple corporate model of investment with follow up continuation,

expansion and contraction options in the presence of ambiguity. We considered two approaches

28

to decision making under ambiguity which derive from two different relaxation of Savage’s SEU

paradigm: the MEU approach, based on the relaxation of the independence axiom, and the

CEU, based on the relaxation of the completeness axiom.

Our point of departure from the literature is in adding CEU preferences to the analysis of real

options. While some recent studies of real options have been studied under MEU preferences, we

are able to compare and contrast MEU with CEU and find significant differences. First, MEU

is indistinguishable from a pessimistic SEU decision maker, offering little hope that empirical

studies will be able to distinguish one from the other. On the other hand, CEU predicts that

management will appear to be optimistic in making decisions to continue a project in the presence

of negative information. This is consistent with the empirical observations that managers are

reluctant to abandon losers and indeed seem to have escalating commitments to projects that

they have started. Second, we propose that CEU be considered as a model of a group of

decision makers, such as a board, a management team or a financing syndicate that makes

decisions based on unanimity. Such a group will be more reluctant to abandon investments as

the group increases in size and hence in effective priors. This interpretation raises the empirical

questions of how groups determine the status quo, what contributes the existence of multiple

priors in a group, and the extent to which groups effectively follow the status quo. Third, we

show that the importance of the status quo in CEU preferences can result in investment decisions

that are time inconsistent. We demonstrate this with a case where an investment that will be

rejected due to an anticipated change in the status quo. That is, when the initial investment is

being considered, the decision maker recognizes that he will be reluctant to abandon a project

once it starts because of the status quo that the initial investment puts in place. As a result,

the initial investment, which does not share this status quo, is not made. Finally, we show

that contract design can change the payoffs in such a case so that the escalating commitment is

neutralized making the initial investment attractive. This raises an entirely new role for financial

contracting, one that anticipates the need to modify payoffs so that future decisions will be time

consistent. In the context of a group, contracts can be shaped so that to induce future unanimity

that is needed for investment to proceed in the first place.

29

A Appendix. Decision theory toolkit

In this appendix we review the theoretical foundations for the three decision decision models used

in the paper, Subjective Expected Utilty (SEU), Minimum Expected Utility (MEU) and Con-

sensus Expected Utility (CEU). Our analysis is cast in the framework developed by Anscombe

and Aumann (1963) and draws heavily on the review article by Gilboa and Marinacci (2011).

A.1 Preliminaries

Let S denote a set describing the possible states of the world, endowed with an event algebra

Σ, Y be a set describing possible outcomes, and ∆(Y ) the space of lotteries over outcomes Y .

A lottery is a random variable with outcomes in Y whose probabilities are objectively known by

the decision maker (DM). For simplicity of exposition, we take S and Y as discrete sets.

The DM makes choices over acts, which are functions that map states into lotteries. Formally,

an act f is a Σ-measurable function f : S → ∆(Y ). Because the space ∆(Y ) is convex one can

construct convex combination of acts, i.e., given any two acts f and g and α ∈ [0, 1] the mixed

act αf + (1− α)g is a function from S to ∆(Y ) defined as

(αf + (1− α)g)(s) = αf(s) + (1− α)g(s), ∀s ∈ S. (A1)

Let F be the space of acts. The DM has preferences � on F , i.e., f � g means that act f is

weakly preferred to g. Note that a preference � over acts induces a preference �∆ over lotteries

if, for all p and q in ∆(Y ) one defines

p �∆ q ⇐⇒ f � g, (A2)

where f and g are constant acts, i.e., f(s) = p and g(s) = q for all s ∈ S. Because constant act

are not subject to state uncertainty, the preference �∆ captures preference with respect to risk,

as opposed to uncertainty.

Example 1. Let us consider a risk neutral manager who has I dollars and considers whether or

not to invest in a project that yields random outcomes. Suppose there are two possible state of

nature ‘up’ and ‘down’. In state ‘up’ the project outcome is yu and in state ‘down’ the outcome

is yd. If the manager does not invest, the outcome is simply I.

30

In terms of the above notation, the state space is S = {down, up} and the outcome space is

Y = {yu, yd, I}. “Investing” is an act f such that:

f(u) =

yu, with prob. 1

yd, with prob. 0

I, with prob. 0

, f(d) =

yu, with prob. 0

yd, with prob. 1

I, with prob. 0

, (A3)

while “Not investing” is an act g such that

g(u) =

yu, with prob. 0

yd, with prob. 0

I, with prob. 1

, g(d) =

yu, with prob. 0

yd, with prob. 0

I, with prob. 1

. (A4)

In other words f and g are degenerate lotteries, which is a formal way to express the fact that

upon investing we know for sure the outcome we obtain in each state, although we do not know

a priori which one of the two states, ‘up’ or ‘down’ will materialize.

In the sequel we summarize the axioms on the primitive preferences � on which the three

model of choice we consider in the paper rest, and provide the representation of the preference

relation �.

A.2 Subjective Expected Utility (SEU)

Savage’s SEU model of decision making rests on the following axioms:

S1. Weak order. � on F is complete and transitive.

S2. Monotonicity. For any f, g ∈ F , if f(s) �∆ g(s) for each s ∈ S, then f � g.

S3. Independence. For any f, g, h ∈ F , and any α ∈ (0, 1) we have

f � g ⇒ αf + (1− α)h � αg + (1− α)h (A5)

S4. Archimedean. Let f, g, h ∈ F be such that f � g � h. Then there are α, β ∈ (0, 1) such

that

αf + (1− α)h � g � βf + (1− α)h (A6)

31

S5. Nondegeneracy. There are f, g ∈ F such that f � g.

The following theorem characterize the SEU representation of preferences.

Theorem 2. Let � be a preference defined on F . The following conditions are equivalent:

1. The preference � satisfies the axioms S1–S5.

2. There exist a non constant function u : Y → R and a unique probability measure π : Σ→

[0, 1] such that for all f, g ∈ F ,

f � g ⇐⇒∑s∈S

∑x∈suppf(s)

u(x)f(s)

π(s) ≥∑s∈S

∑x∈suppf(s)

u(x)g(s)

π(s) (A7)

In the case of the risk neutral manager considered in Example 1, the two acts: invest, f

and don’t invest, g, map into degenerate lotteries on the outcome space and hence the above

theorem implies that

Invest � Don’t invest ⇐⇒ Eπ[x̃] ≥ I, (A8)

where Eπ[u(x̃)] = π(u)yu + π(d)yd.

A.3 Minimum Expected Utiltiy (MEU)

Gilboa and Schmeidler (1989) start from Savage axioms and:

i. Replace the independence axiom S3 over acts with the following independence axiom over

lotteries

M3. C-Independence. For all acts f, g ∈ F and all constant acts (lotteries) p

f � g ⇒ αf + (1− α)p � αg + (1− α)p, ∀α ∈ [0, 1] (A9)

ii. Introduce a new axiom to capture aversion to uncertainty

M6. Uncertainty Aversion. For any f, g ∈ F and any α ∈ (0, 1)

f ∼ g ⇒ αf + (1− α)g ≥ f. (A10)

32

The following theorem characterize the MEU representation of preferences.

Theorem 3. Let � be a preference defined on F . The following conditions are equivalent:

1. The preference � satisfies the axioms S1, S2, M3, S4, S5, M6.

2. There exist a non constant function u : Y → R and a convex and compact set Π ⊆ ∆(Σ)

of probability measures such that for all f, g ∈ F ,

f � g ⇐⇒ minπ∈Π

∑s∈S

∑x∈suppf(s)

u(x)f(s)

π(s) ≥ minπ∈Π

∑s∈S

∑x∈suppf(s)

u(x)g(s)

π(s)

(A11)

In the case of the risk neutral manager considered in Example 1 the above theorem implies

that

Invest � Don’t invest ⇐⇒ minπ∈Π

Eπ[x̃] ≥ I, (A12)

where Eπ[u(x̃)] = π(u)yu + π(d)yd.

A.4 Consensus Expected Utiltiy (CEU)

Bewley (2002) start from Savage’s axioms and replaces the completeness axiom over acts with

a completeness axiom over lotteries. Formally,

C1. C-Completeness. For every constant act (lottery) p, q ∈ ∆(Y ), p � q or q � p.

The following theorem characterize the CEU representation of preferences.18

Theorem 4. Let � be a preference defined on F . The following conditions are equivalent:

1. The preference � satisfies the axioms C1 and S2–S5.

2. There exist a non constant function u : Y → R and a convex and compact set Π ⊆ ∆(Σ)

of probability measures such that for all f, g ∈ F ,

f � g ⇐⇒∑s∈S

∑x∈suppf(s)

u(x)f(s)

π(s) ≥∑s∈S

∑x∈suppf(s)

u(x)g(s)

π(s), ∀π ∈ Π.

(A13)18The theorem is based on the characterization provided by Gilboa, Maccheroni, Marinacci, and Schmeidler

(2010) who represent weak preferences. Bewley (2002) provides characterization of strict preferences.

33

In the case of the risk neutral manager considered in Example 1 the above theorem implies

that

Invest � Don’t invest ⇐⇒ Eπ[u(x̃)] ≥ I, ∀π ∈ Π, (A14)

where Eπ[x̃] = π(u)yu + π(d)yd.

B Appendix. Proofs

Proof of Proposition 1

Consider SEU preferences and suppose there exists a plan ℘′ = (a′u, a′d) ∈ P

∗SEU such that ℘′ � q

for a given plan q ∈ P and that q � ℘ = (au, ad) for all ℘ ∈ P∗SEU. We will now construct a

maximal plan in the set of implementable plan: Because A is finite, for each node j, there must

be an action a′′j such that

Ep[C̃a′′j (s̃j2)] = sup

a∈AEp[C̃a(s̃

j2)].

By construction, the plan ℘′′ = (a′′u, a′′d) ∈ P∗SEU and satisfies

Ep(C̃℘′′) > Ep(C̃℘

′) > Ep(C̃q).

As a result, ℘′′ is preferred to q. This is a contradiction because ℘′′ ∈ P∗SEU and so it cannot be

that it is prefered to q.

Consider MEU preferences and suppose there exists a plan ℘′ = (a′u, a′d) ∈ P

∗MEU such that

℘′ � q for a given plan q ∈ P and that q � ℘ = (au, ad) for all ℘ ∈ P∗MEU. The procedure is

similar for MEU and we will construct similarly a maximal plan in the set of implementable

plan: For each node j, there must be an action a′′j such that

minπ∈Π

Eπ[C̃a′′j (s̃j2)] = sup

a∈Aminπ∈Π

Eπ[C̃a(s̃j2)]. (B15)

34

By construction, the plan ℘′′ = (a′′u, a′′d) is implementable. Let us now prove that the plan ℘′′ is

prefered to q by the MEU DM:

minπ,πu,πd∈Π3

Eπ,πu,πd(C̃℘

′′) = min

π,πu,πd∈Π3

[πEπu [C̃a

′′u(s̃u2)] + (1− π)Eπd

[C̃a′′d (s̃d2)]

](B16)

= minπ∈Π

[π minπu∈Π

Eπu [C̃a′′u(s̃u2)] + (1− π) min

πd∈ΠEπd

[C̃a′′d (s̃d2)]

](B17)

> minπ∈Π

[π minπu∈Π

Eπu [C̃a′u(s̃u2)] + (1− π) min

πd∈ΠEπd

[C̃a′d(s̃d2)]

](B18)

= minπ,πu,πd∈Π3

Eπ,πu,πd(C̃℘

′) > min

π,πu,πd∈Π3Eπ,πu,πd

(C̃q) (B19)

and as a result, ℘′′ is preferred to q. This is a contradiction because ℘′′ ∈ P∗MEU and so it cannot

be that it is prefered to q. Notice that equality (B16) is just the definition of the utility derived

by the MEU DM. Equality (B17) follows from the fact that π and 1−π are non negative and so

we can optimize sequentially over the triplet (π, πu, πd). Equation (B18) is implied by the fact

that ℘′′ is maximal in the set of implemtable plan (e.g. equation (B15)). The equality is line

(B19) is the definition of the derived utility from the outcome of the action ℘′ and the inequality

follows from the fact that we assumed that ℘′ � q.

Proof of Proposition 2

Let us start with a the following two lemmas:

Lemma B.1. Suppose Assumption 1 and conditions (29) and (32) are satisfied. If the en-

trepreneur issues a convertible bond with conversion ratio α ∈ (0, 1) and face value X satisfying

condition (33) and undertakes the investment at time t = 0, the entrepreneur’s option exercise

choice at time time t = 1 in the ‘up’ state is to expand the firm and the bondholder’s conversion

choice is to convert the bond is all state at time t = 2.

Proof: When the entrepreneur reaches the node up at time t = 1, the subsequent firms cash

flow C̃2 and payoff to investors and the entrepreneur can be described for each real investment

choice at time t = 1:

• If the entrepreneur decides to expand the firm, then the firm cas flows are C̃2 = λS̃u2 − I1.

The investors will have to decide whether to convert the bond or take its face value. It is

35

easy to check that conditions (32) and (33) imply

X < αR < α(λsud − I1) < α(λsuu − I1)

and the financier will choose to convert the bond in both states ‘up-up’ and ‘up-down’.

Consequently, if the entrepreneur chooses to expand the firm, he will be left with the cash

flow (1− α)(λs̃u2 − I1) as time t = 2

• If the entrepreneur chooses to continue the firm (rather than expanding or shutting down

the firm), the firm cash flows are given by C̃2 = s̃u2 . Because R < s̃u2 and that condition

(33) implies that X < αR, the investor will convert the bond and the entrepreneur is left

with (1− α)s̃u2 .

• If the entrepreneur chooses to shut down the firm, the firm cash flow is C̃2 = R and the

share αR goes to the financier whereas the share (1− α)R goes to the entrepreneur.

To summarize, for all option exercise policies, the entrepreneur’s cash flow subsequent to

the node ‘up’ is given by (1 − α)C̃2. When the convertible is issued, the entrepreneur’s pay-

off subsequent to node ‘up’ is simply a scaled version of the entrepreneur’s payoff when the

convertible is not issued. Consequently, the entrepreneur’s incentives are not distorted in node

‘up’ and the choice of option exercise policy is identical to the choice that the entrepreneur will

undertake when the convertible is not issued. When the convertible is not issued, recall that

using conditions (12) and (13) together with Assumption 1 shows that the CEU entrepreneur

will expand the firm at node ‘up’. The entrepreneur will thus expand in the the ‘up’ state in

presence on a convertible bond. This concludes the proof of Lemma B.1.

Lemma B.2. Suppose Assumption 1 and conditions (29) and (32) are satisfied. If the en-

trepreneur issues a convertible bond with conversion ratio α ∈ (0, 1) and face value X satisfying

condition (33) and undertake the investment at time t = 0, the entrepreneur’s option exercise

choice at time time t = 1 in the ‘down’ state is to shut down the firm and the investors takes a

proportion α of the scrap value R .

Proof: When the entrepreneur reaches the node ‘down’ at time t = 1, the subsequent firms cash

flow C̃2 and payoff to investors and the entrepreneur can be described for each real investment

choice at time t = 1:

36

• If the entrepreneur expands the firm at node ‘down’, the firm’s cash flow is C̃2 = λs̃d2 − I1

at t = 2. The financier will then receive the cash flow ϕ(λs̃d2 − I1) where the function ϕ

summarizes the convertible exercise policy of the investor and is defined by

ϕ(y) = max{αy,min{X, y}}. (B20)

The entrepreneur ends up then with the cash flow

ξ̃E expands in ‘down’

2 = ψ(λs̃d2 − I1)

where the function ψ defined by

ψ(y) = y − ϕ(y).

describes the payoff to the entrepreneur when the bond is optimally exercised by the

investor. Notice that the function ψ is non-decreasing because α < 1.

• If the entrepreneur decides to continue the firm at node ‘down’, the firms cash flow is

C̃2 = s̃d2 at time t = 2. Assumption 1 and condition (33) imply that αsdd < X < αR < αsdu

and therefore the the investor will exercise the conversion option at node ‘down-up’ but

not at node ‘down-down’. This yield the bondholder’s payoff

B̃E continues in d2 =

{αsdu if s̃d2 = sdu

X if s̃d2 = sdd(B21)

and the resulting payoff to the entrepreneur

ξ̃E continues in ‘down’2 =

{(1− α)sdu if s̃d2 = sdu

sdd −X if s̃d2 = sdd(B22)

• If the entrepreneur decides to contract, the bond holders get αR and the entrepreneur gets

(1− α)R.

First let us show that when facing the binary decision to expand relative to the status quo

(continue the firm) at time t = 1 in the ‘down’ state, the entrepreneur does not expand and

chooses the status quo. The difference is cash flow to the entrepreneur in state ‘down-up’ satisfies

37

[ξ̃E continues in ‘down’

2 − ξ̃E expandss in ‘down’

2

]du

= (1− α)sdu − ψ(λsdu − I1)

≥ (1− α)sdu − ψ(sdu) (B23)

= ϕ(sdu)− αsdu ≥ 0. (B24)

Condition (32)implies λsdu − I1 < sdu and since ψ is monotonic we get inequality (B23)). The

last inequality in (B24) comes from the definition of ϕ given in (B20).

Similarly, in the ‘down-down’ state, the difference in cash flow to the entrepreneur is

[ξ̃E continues in ‘down’

2 − ξ̃E expandss in ‘down’

2

]dd

= sdd −X − ψ(λsdd − I1)

≥ sdd −X − ψ(sdd) (B25)

= ϕ(sdd)−X ≥ 0. (B26)

Condition (32)implies λsdd − I1 < sdd and since ψ is monotonic we get inequality (B25)). The

last inequality in (B26) comes from the definition of ϕ given in (B20) and from the fact that

condition (33) implies X < sdd.

To summarize, when the firm is started after the convertible is issued, the entrepreneur’s

payoff resulting from expanding the firm at node ‘down’ is dominated in all states subsequent

to node ‘down’ by the payoff resulting from continuing the firm. Consequently, at time t = 1,

the CEU entrepreneur will not expand the firm in the ‘down state’.

Second, let us show that facing the binary decision to shut down the firm relative to the

status quo (continue the firm) at time t = 1 in the ‘down’ state, the entrepreneur does not

choose the status quo and rather shut down the firm.

Using the payoff analyzes under different real investment policies, it is useful to write the

payoff to the entrepreneur if he shut down the firm in the ‘down’ state as

ξ̃E continues in ‘down’2 = (1− α)s̃d2 − (X − αsdd)1‘down−down‘

38

where 1‘down−down‘ is an indicator random variable equal to 1 if state ‘down-down’ occurs and

zero otherwise. Using this formula, it is easy to see that the entrepreneur will shut down the

firm at time t = 1 in the ‘down’ state if and only if

(1− α)R > (1− α)Eπ[sd2]− (1− π)(X − αsdd), ∀π ∈ Π, (B27)

which is equivalent to

(1− α)R > (1− α)Ep+ε[s̃d2]− (1− p− ε)(X − αsdd), (B28)

or,

X > αsdd +(1− α)Ep+ε[s̃d2]−R

1− p− ε. (B29)

The last inequality is stated in condition (33) and therefore, due to the presence of the

convertible bond, the entrepreneur chooses to shut down the firm in the ‘down‘ state which in

turn concludes the proof of Lemma B.2.

Now let us turn to the proof of Proposition 2. At time t = 0 contemplates starting the firm

after issuing the convertible or remain with the status quo I0. The results in Lemma B.1 and B.2

show that the anticipated real investment policy is the one in which the entrepreneur expands

in state ‘up’ and contract in state ‘down’. Therefore, the convertible bond is a gamble b̃ with

time t = 2 payoffs α(λs̃u2 − I1) in states ‘up-up’ and ‘up-down‘ and αR in the states ‘down-up’

and ‘down-down’. It follows that a CEU financier purchasing such a convertible bond will be

willing to pay a price P such that

b̃− P �CEU 0 ⇐⇒ P < α [(p− ε) (λEp−ε[s̃u2 ]− I1) + (1− p− ε)R] . (B30)

Notice that the price of the bond is independent of the face value X. The face value only

plays the role of a “threat” to induce the entrepreneur to avoid taking actions that will result in

having to pay X to the bondholder. The entrepreneur undertakes the investment at time t = 0

if

I0 − P < (1− α) [(p− ε) (λEp−ε[s̃u2 ]− I1) + (1− p− ε)R] . (B31)

39

Using (B30), inequality (B31) is equivalent to

I0 < (p− ε)(λEp−ε[s̃u2 ]− I1) + (1− p− ε)R, (B32)

When condition (29) is satisfied, conditions (B32) holds and as a result the entrepreneur issues

the convertible and starts the project. This concludes the proof of Proposition 2.

C Appendix: Precommitment via securities with action-dependent

payoff

We explore in this section some alternative contracts solving the time inconsistency problem of

the CEU entrepreneur. By allowing the issued securities to have payoffs that are contingent on

the entrepreneur’s actions, we are able to target more directly the time inconsistency problem.

While these securities are not always realistic, the exercise helps understand what contractual

features of the more realistic convertible bond are effective in providing a solution to the time

inconsistency problem.

The objective of the entrepreneur here is to render the investment plan (E ,R) implementable.

Referring to Panel B of Figure 4, the entrepreneur would like to displace the point R outside

the shaded area so that it is preferred to the status quo. This can be achieved either by altering

the payoff resulting from the contraction (i.e. displacing R) or by altering the payoff associated

the status quo (i.e. displacing fd). Following the framework of Proposition 2 , we suppose that

Assumption 1 and conditions (29) and (32) are satisfied.

Let us suppose the entrepreneur acquires at t = 0, for a cost Pg, a security g at time t = 0

that pays off η > 0 if he shuts down the firm and zero otherwise. This security has a ‘carrot-like’

feature. Such a security will move upward the point R along the 45 degree line in both panels

A and B of of Figure 4 and if

Ep+ε[s̃d2] < R+ η < Ep+ε[s̃u2 ], (C33)

the entrepreneur will decide to abandon the status quo in the down state and shut down the

firm (first inequality in (C33)). The second inequality (C33) guarantees that η is not too large

to induce the entrepreneur to shut down also in the up state.

40

Under Condition (C33), the investment plan (E ,R) becomes implementable. At time t = 0

E will invest in the project if

I0 + Pg < (p− ε)(λEp−ε[s̃u2 ]− I1) + (1− p− ε)(R+ η). (C34)

If the financier who sells the security to the entrepreneur also has CEU preferences with the

same prior set Π as the entrepreneur,19 the security will be priced at least for

Pg = (1− p− ε)η (C35)

provided the project is undertaken by the entrepreneur. This is because the financier expects the

entrepreneur to shut down the firm only in state ‘d’ and therefore the security g is a gamble with

payoff equal to η in states ‘down-up’ and ‘down-down’ and zero otherwise. Plugging equation

(C35) in (C34) shows that the entrepreneur undertakes the project if and only if

I0 < (p− ε)(λEp−ε[s̃u2 ]− I1) + (1− p− ε)R. (C36)

The right hand side of (C36) is exactly the payoff of the plan (E ,R) which we assumed to be

strictly preferred to the status quo I0 (see condition ((29))) Therefore, the entrepreneur will buy

the security g that will induce him to shut down the firm in the ‘d’ state and expand the firm

in the ‘u’ state, and as a result invest in the project. By purchasing the security g that pays off

η when the firm is shut down the entrepreneur solves the time inconsistency problem.

A similar outcome can be obtained via a “stick-like” security. Suppose instead that the

entrepreneur issues at time t = 0 a security h according to which he promises to pay an amount

µ > 0 in case the firm is not shut down. We can think of the payoff required by h as of a sort

of maintenance cost. In panel A and B of Figure 4 this security will leave the point R in the

same position but moves down the points f and λf − I along the 45 degree line. To induce the

entrepreneur to shut down the firm in the ‘d’ state without altering the decision to expand in

the ‘u’ state, the payoff µ has to be such that

Ep+ε[s̃d2]− µ < R < Ep+ε[s̃u2 ]− µ. (C37)

19The analysis will be qualitatively similar if we were to assume a financier with SEU or MEU preferences.

41

When Condition (C37) is satisfied, the investment plan (E ,R) becomes implementable. The

entrepreneur will undertake the project at time t = 0 if

I0 − Ph < (p− ε)(λEp−ε[s̃u2 ]− I1 − µ) + (1− p− ε)R. (C38)

where Ph is the price that the entrepreneur charges for the security h. Because the entrepreneur

will shut down the firm in the state ‘d’, the security will be priced as a security that pays µ in

state ‘u’ and zero otherwise. If the financier also has CEU preferences with the same prior set

Π as the entrepreneur, the price of the security h should be lower than

Ph = (p− ε)µ, (C39)

Plugging equation (C39) into (C38) shows that the entrepreneur will invest in the project

when condition (29) holds. To summarize, the entrepreneur receives the price Ph to issue a

security h that will induce him to shut down in the ‘d’ state and as a result, he will invest in

the project.

42

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