Repeated Signaling and Firm Dynamics∗
Christopher A. Hennessy
Dmitry Livdan
Bruno Miranda
First Draft: April 25, 2006
Current Draft: October 9, 2007
Abstract
We examine the effect of repeated (Markovian) hidden information on the dynamics of financing and
investment. The model features endogenous investment, debt, default, dividends, equity flotations and
share repurchases. The privately informed principal has an endogenous precautionary motive to hoard
cash and capital in order to avoid future signaling costs. Consequently, in the least-cost separating
equilibrium, firms with negative information have no debt and overinvest relative to first-best. Sufficient
conditions for single-crossing are established. Firms credibly signal positive information by substituting
debt for equity. Default costs on the debt induce underinvestment relative to first-best. The need for
costly signals is decreasing in net worth. We show the equilibrium set includes pooling equilibria if and
only if net worth is sufficiently low. Using simulated method of moments to estimate model parameters,
estimated default costs are 21% of firm value. The calibrated model is consistent with a broad set of
stylized facts regarding investment, leverage, payouts, and announcement effects.
1
Introduction
There is now general consensus that imperfect information has important implications for the dynamics of
firm investment and financing. Existing dynamic models focus on hidden action. For example, Bernanke
and Gertler (1989) and Carlstrom and Fuerst (1997) show that moral hazard helps explain pronounced
investment cyclicality. Clementi and Hopenhayn (2006), DeMarzo and Fishman (2007) and Biais et al.
(2007) show empirically observed dynamics of leverage and dividends can be explained by contracting under
moral hazard.
In this paper, we examine the effect of hidden information on firm investment and financing by developing
a dynamic signaling model. Consistent with empirical evidence, the signaling model developed herein predicts
∗ The authors are from U.C. Berkeley, U.C. Berkeley, and Indymac Bank, respectively. We thank seminar participants
at Columbia, Dartmouth College, NYU, LBS, LSE, Bank of England, HEC Lausanne, Minnesota, University of Vienna,
Washington University, Stanford (SITE), SED (Prague), CEPR (Norway), the 2006 Lone Star Conference, the 2007 WFA
Meetings, and the Hansen-Singleton Conference at CMU. We owe special thanks to Sudipto Bhattacharya, Patrick Bolton,
Peter DeMarzo, Andrea Eisfeldt, Jean Charles Rochet, Neng Wang, and Toni Whited. This paper was previously circulated
under a different title.
1
that investment and dividends will increase with net worth, whereas leverage ratios will decrease with net
worth. Additionally, signaling explains another set of stylized facts that moral hazard does not: abnormal
equity returns specifically connected to announced corporate policies.1
There is, in fact, considerable
empirical evidence that financial markets interpret policy choices as signaling insider information.2
Another contribution of this paper is methodological. Maskin and Tirole (1992) develop a general
algorithm for computing the set of perfect Bayesian equilibria (PBE) in static signaling games. Tirole
(2006) utilizes this static framework to analyze security issuance by a privately informed insider. In this
paper, we develop a tractable recursive approach for computing PBE for infinitely-lived public firms. Our
treatment differs from the static model of Tirole (2006) in four important respects. First, the equilibrium
set of each periodic signaling game changes as net worth evolves endogenously. Second, Tirole considers
exogenous utility functions for the informed principal and uninformed agent. In our model, utilities depend
on endogenous equity value functions that capitalize the payoffs in future signaling games. Third, in a static
model, default occurs when debt exceeds internal funds. In our dynamic model, future cash flow can be
pledged to outside investors, increasing debt capacity and reducing default risk. Finally, in a static setting,
the firm necessarily distributes all cash flow. In our dynamic model, cash retentions are critical to reducing
losses due to asymmetric information.
A related methodological contribution of the paper is that it offers a tractable quantitative framework for
analyzing the effects of hidden information. The model can be taken directly to the data since investment,
debt, equity flotations, dividends, share repurchases, and default are all endogenous. Gomes (2001), Cooley
and Quadrini (2001), and Hennessy and Whited (2007) develop calibrated models incorporating direct
flotation costs. Carlstrom and Fuerst (1997) develop a calibrated general equilibrium model with moral
hazard. In this paper, we use simulated method of moments in order to estimate deep structural parameters,
such as the driving process for hidden information.
We start with a standard neoclassical investment framework where the firm has a concave profit function
and default entails proportional deadweight losses. There is a Markovian profit shock privately observed by
the controlling insider-shareholder at the start of each period. Outside investors have inferior information,
observing each realized profit shock with a one-period lag. Using his private information, the insider
maximizes the value of his equity stake. The insider is forward-looking and recognizes that current policies
will affect the severity of future adverse selection problems.
In the baseline model, attention is confined to least-cost separating equilibria. In this setting, anticipation
of future signaling costs causes the insider to behave as if risk-averse, with his utility a concave transformation
of net worth.3 The intuition is as follows. Suppose the insider has positive information, but net worth is low.
The insider would need to raise a large amount of external funds in order to implement first-best investment.
1 In moral hazard models, only earnings announcements induce abnormal returns. After earnings and net worth are revealed,
policies can be inferred and announcements of policies have no effect on equity prices.
2 See Eckbo, Masulis and Norli (2007) for a comprehensive survey of event studies on security issuance.
3 Clementi and Hopenhayn (2006), Biais et al. (2007) and DeMarzo and Fishman (2007) show that repeated hidden action
generates concave continuation values.
2
However, the issuance of a large block of equity would create a strong temptation for firms with negative
information to mimic, since they stand to gain a great deal from security mispricing. In order to separate,
the insider with positive information must substitute risky debt for equity. Consequently, the marginal
value of internal funds for this firm exceeds one, since internal funds reduce the need for external funds and
concomitant signaling costs. On the other hand, a firm with high net worth does not need external funds
to finance desired investment. Such a firm will distribute a marginal dollar as a dividend. Therefore, the
shadow value of internal funds is equal to one for high net worth firms.
Least-cost separating contracts minimize the temptation of insiders with negative information to mimic.
Since insiders are pseudo-risk-averse, investors provide those reporting low expected profitability with partial
insurance against future signaling costs. In particular, low types are financed only with equity, hold cash
buffer stocks, and overinvest relative to first-best. These predictions are consistent with the empirical findings
of Graham (2000) and Leary and Roberts (2006), who document debt conservatism and equity-first financing
by a subset of firms.
Sufficient conditions for single-crossing in debt-equity and capital-equity space are established. Under
these conditions, firms with positive information and low net worth finance with debt. Default costs reduce
the return to capital accumulation, resulting in underinvestment relative to first-best. As net worth increases,
so too does the investment of firms with positive private information. Thus, the model provides a rigorous
microfoundation for the informal argument of Fazzari, Hubbard and Petersen (1988) that hidden information
induces a positive propensity to invest out of cash windfalls.
In simulated data, the leverage ratio declines with lagged profits. This is because low lagged profits lead
to low net worth and large financing gaps. In such states, simulated leverage ratios are particularly high
since high types must send strong signals if they are to avoid adverse selection associated with the flotation
of a large block of securities. The same causal mechanism reconciles the model with observed cyclicality
in leverage ratios. In the model, there are private and public shocks to profits, with the latter proxying
for commonly observed macroeconomic conditions. Following low realizations of the public macroeconomic
shock, firms have low net worth. In such states, simulated leverage ratios are particularly high. These
predictions are consistent with stylized facts regarding leverage dynamics documented by Fama and French
(2002) and Korajczyk and Levy (2004).
In line with the empirical findings of DeAngelo, DeAngelo and Stulz (2006) dividends are initiated only
when net worth is sufficiently high. In the model, firms with negative private information initiate dividends
when the marginal precautionary value of internal funds is just equal to the marginal cost of retentions.
Firms with positive private information only initiate dividends when the incentive constraint is slack. Since
the incentive of the low type to mimic depends on his gain from security mispricing, the incentive constraint
becomes slack only when the high type has sufficient net worth such that investment is primarily financed
with internal funds.
The estimated model generates predictions consistent with observed patterns of abnormal equity returns
3
surrounding corporate announcements. Simulated share prices exhibit positive abnormal returns in response
to high capital expenditures and high leverage. This is consistent with the findings of McConnell and
Muscarella (1985) and Masulis (1983).
We also extend the baseline model to consider the existence of alternative equilibria. Maskin and Tirole
(1992) show the set of PBE in a static signaling game consists of allocations Pareto dominating the leastcost separating contract. The equilibrium set in our model varies with net worth. When net worth is low,
high types must incur large signaling costs in order to achieve separation. Thus, pooling equilibria exist if
net worth is sufficiently low. In the ex ante optimal pooling equilibrium, neither type finances with debt,
since debt is costly. In this pooling equilibrium, firms with positive (negative) private information regarding
growth options underinvest (overinvest) relative to first-best. In addition to net worth, the lagged type also
influences the equilibrium set. In particular, persistence in types implies that investors have favorable priors
for firms that were profitable in the preceding period. Since favorable priors encourage pooling, firms with
high lagged profits pool over a wider range of net worth levels.
The model is related to a number of papers that analyze static signaling. Spence (1973) pioneered this
literature, with Leland and Pyle (1977), Ross (1977) and Bhattacharya (1979) the first to analyze financial
signals. Quinzii and Rochet (1985) and Milgrom and Roberts (1986) analyze multi-dimensional signaling.
Ambarish, John and Williams (1987) and Williams (1988) analyze multi-dimensional financial signals. There
is no debt in these models. In contrast, Viswanathan (1995) allows signaling through debt and investment in
a static setting where asymmetric information only concerns assets-in-place. In addition to being dynamic,
our model accommodates asymmetric information regarding growth options and assets-in-place.
In terms of static models, our model is most closely related to that of Constantinides and Grundy (1990).
They prove a firm with variable investment scale can implement first-best if bankruptcy is costless. First-best
is achieved with the firm issuing debt in excess of the amount needed to fund investment, with excess funds
used for share repurchases. Aside from introducing dynamics, we depart from Constantinides and Grundy
by introducing default costs. This precludes first-best investment. In fact, default costs are responsible for
high types’ underinvestment in our model. Endogenous precautionary motives are responsible for low types’
overinvestment in our model.
Nachman and Noe (1994) assume investment scale is fixed and confine attention to securities with payoffs
that increase monotonically with cash flow.4 Under these assumptions, there is no separating equilibrium,
as firms always benefit from reporting the highest type and receiving the highest security value (due to
monotonicity). Nachman and Noe derive sufficient conditions such that debt is the unique contract. The
essential idea is that debt minimizes cross-subsidies. The central difference between our setting and that
considered by Nachman and Noe is that we allow the firm to signal using investment scale and share
repurchases. The broader, and arguably more realistic, signal space creates the possibility for separating
equilibria.
4 DeMarzo and Duffie (1999) obtain similar results under rather different timing assumptions. They analyze security design
by an uninformed principal who will become informed prior to selling securities in secondary markets.
4
There is limited existing work modeling investment and financing under repeated hidden information.
Lucas and McDonald (1990) develop a dynamic model of investment under hidden information. However,
they constrain the firm to finance with equity. Sannikov (2006) presents a model of optimal security design
when there is one-time hidden information ex ante and repeated moral hazard ex post.
The remainder of this paper is organized as follows. Section 2 describes the economic setting. Section 3
characterizes separating equilibria qualitatively. Section 4 presents model estimation and simulation. Section
5 discusses pooling equilibria. Section 6 concludes.
2
The Economic Environment
2.1
Technology
Time is discrete and the firm’s horizon is infinite. There is a riskless asset paying interest rate r > 0. All
parties are risk-neutral and share the discount factor β ≡ (1 + r)−1 . Physical capital (k) decays exponentially
at rate δ ∈ [0, 1]. We impose standard restrictions on the firm’s profit function.
Assumption 1. The operating profit function π : K × Θ × E → <+ has the following properties: (i) weakly
positive; (ii) strictly increasing; (iii) strictly concave in capital; (iv) twice continuously differentiable; and
(iv) satisfies the Inada conditions.
The price of capital is one. Following Cooley and Quadrini (2001), there are no costs to adjusting the
capital stock.
Assumption 2. The physical capital stock is perfectly reversible.
Assumption 2 limits the number of state variables. Since capital is perfectly reversible, it may be added to
cash to compute fungible net worth (wt ).
The driving processes are described next. For reasons made obvious below, θt is labeled a private shock
and εt a public shock.
Assumption 3. The private shocks {θt }∞
t=1 take values in Θ ≡ {θ L , θ H } and follow a first-order Markov
process with no absorbing state. The public shocks {εt }∞
t=1 are independently and identically distributed with
a continuously differentiable density function f : [ε, ∞) → [0, 1] satisfying f (ε) > 0 for all ε ≥ ε > 0.
Let p(θi |θj ) denote the probability of θi conditional on lagged type θj .
The firm can finance its activities using internal resources, external equity, or debt. Funds may be
distributed to shareholders using dividends or share repurchases. In distinguishing between alternative modes
of distribution, the model is unique within the class of dynamic structural models. Dividends (d) cannot be
5
negative.5 The face value of debt is denoted b and the price of debt is denoted ρ. The borrowing technology
is similar to that assumed by Cooley and Quadrini (2001). However, the method used to determine the
default threshold is different. Cooley and Quadrini (2001) allow shareholders of a distressed firm to directly
inject their own funds (at a cost) to meet an outstanding debt obligation. In our model, funds can only
be raised from bona fide uninformed outside investors. The next assumption summarizes the contracting
space.6
Assumption 4. The contract space consists of equity and single-period debt. Default occurs if the firm
cannot raise sufficient funds in its next financing round to meet an outstanding debt obligation. Lenders are
senior in default and incur proportional (φ > 0) bankruptcy costs.
Following Carlstrom and Fuerst (1997), Gomes, Yaron and Zhang (2003) and Cooley, Marimon and
Quadrini (2004), the firm faces the risk of a catastrophic event (exponential death). As concrete examples, one
may think of the catastrophic event as approximating mass tort claims for defective products or expropriation
by a government.
Assumption 5. Each period there is a probability 1 − γ of a catastrophic event, where γ ∈ (0, 1). If a
catastrophic event occurs, all marketable claims on the firm are worthless.
As noted by Carlstrom and Fuerst (1997), an infinitely-lived firm facing an imperfect financial market
will save all funds provided it can earn the same rate of return as investors. In such a setting, the effect
of financial frictions vanishes quickly as the firm saves its way out of the problem. However, the possibility
of a catastrophic event provides a countervailing cost to saving within the corporate shell.7 Although it
induces bounded saving, Assumption 5 stands in contrast to existing models that apply a higher discount
rate in equity markets than in debt markets, creating an exogenous bias against equity as a source of external
finance. Assumption 5 ensures debt and external equity face the same catastrophic risk and the same effective
discount factor (βγ).
2.2
Issuance Games
In terms of the real and financial technologies available to the firm, the model is a standard neoclassical
investment framework. Our point of departure is the introduction of hidden information.
Assumption 6. The shock θt is privately observed by a controlling insider-shareholder at the start of period
t. At the end of period t, the realized values of εt and net worth are observed simultaneously by all agents.
5 A negative dividend corresponds to a rights-issue. A rights-issue would be the preferred source of external funding given
asymmetric information. One can rationalize the dividend constraint by appealing to shareholder wealth constraints outside
the model.
6 As a technical assumption, θ is nonverifiable in a court. Otherwise, a θ-contingent debt contract would dominate standard
debt.
7 Corporate income taxes also discourage saving within the firm. However, the introduction of a corporate income tax would
greatly complicate the analysis and limit our ability to compare results with other models in this literature which uniformly
assume exponential death.
6
Assumption 6 implies the insider has a one-period information lead relative to outsiders each period. To
see this, note that an outsider can infer θt after observing net worth and the public shock at the end of
period t. Since any information advantage is short-lived, truthful revelation is here easier to achieve than
under fixed types. The second part of Assumption 6 (observability of wt and εt ), which allows outsiders to
infer the lagged type is not essential. The model encompasses i.i.d. types as a special case. If types are i.i.d.,
one may assume outsiders only observe net worth and the results stated below remain valid.
Figure 1 provides a timeline. There is an infinite sequence of issuance games played between the insider
and atomistic (competitive) outside investors. There are two state variables. The firm enters the period t
financing round with net worth wt and his lagged type, say θj . Both state variables are common knowledge
given Assumption 6. After observing the current type, say θi , the insider offers an option contract to
outsiders. If accepted, the option contract gives the insider the right to choose between two allocations, say
(aLj , aHj ). Equivalently, one can think of the allocation being based on an announced type. An allocation
is a vector determining financing and investment.8 We define aij to be the allocation received by an insider
reporting type i with observed lagged type j. Without loss of generality, attention can be restricted to
contracts inducing truthful reporting of the current type. Allowing allocations to be predicated upon the
lagged type, which is common knowledge at the time of contracting, is important since the lagged type
determines transition probabilities and investor priors. After receiving the option contract offer, outsiders
update beliefs and accept or reject. If the offer is rejected, the insider cannot transact in the firm’s securities.
If the option contract is accepted, the insider is free to implement either aLj or aHj .9 Finally, the firm is
exposed to the catastrophic risk.
At the start of period t + 1, εt is commonly observed. Lenders then determine whether the firm will be
able to deliver the debt obligation bt coming due. In order to make this determination, lenders compute
provisional net worth. The provisional net worth of a type-i that took the type-j allocation is
w(b
e j , kj , θi , ε) ≡ (1 − δ)kj + π(kj , θi , ε) − bj .
There is a cutoff wid < 0 such that a firm of type i can (and will) deliver the promised debt payment if and
only if w
e ≥ wid . It is worth stressing that the default threshold wid depends on the actual current type, which
the lender is able to infer at the time of the default determination. Under the stated timing assumptions
(Figure 1), the default threshold cannot possibly depend on the next realized type since that information is
not yet available to any party at the time of the default determination.
This formulation allows the firm to pay the debt coming due (bt ) using a portion of the proceeds from the
flotation of new securities in financing round t + 1. To see this, note that for any w
e ∈ [wid , 0), the firm has
insufficient internal resources to deliver bt and must therefore use new external financing to cover the debt.
If w
e < wid , lenders know the firm cannot possibly deliver the promised debt payment. In this case, lenders
8 Tirole
(2006) uses a different definition of an allocation.
the parties to the option contract are entering into a direct revelation mechanism.
9 Effectively,
7
are forced to incur bankruptcy costs in order to reset the debt payment.10 After incurring bankruptcy costs,
lenders reset the debt payment to the maximum amount consistent with shareholders’ limited liability. This
bankruptcy process leaves a defaulting firm to enter the upcoming financing round with net worth wid . As
verified below, resetting the debt payment in this manner leaves shareholders of a defaulting firm with a
claim worth zero. The resulting law of motion for net worth is
w(b, k, ε, θi ) ≡ max{wid , w(b,
e k, ε, θi )}.
(1)
After the net worth wt+1 is determined, the insider privately observes θt+1 and the next issuance game
begins.
Insider objectives, stated as Assumption 7, are identical to those adopted by Constantinides and Grundy
(1990).
Assumption 7. The choice of contracts is made by a risk-neutral infinitely-lived insider who maximizes the
expected discounted value of the future dividends coming from fixed stock holdings.
The insider holds m > 0 shares of stock. Total shares outstanding at the start of the period is c. Shares
are issued and repurchased ex dividend. The number of new shares issued is n, with n < 0 indicating a share
repurchase. Let s ≡ n/(c + n) represent the percentage equity stake sold (repurchased). The insider receives
a fraction m/c of total dividends and holds an equity stake of m/(c + n) at the end of the period.
There are (only) two unknown value functions (VL , VH ), with Vj : [wjd , ∞) → <+ . Here Vθt (wt+1 ) denotes
the total value of shareholders’ equity for a firm that realized type θt in period t and entering the t+1 financing
round with net worth wt+1 . It is important to stress that Vθt (wt+1 ) is computed after the lender makes the
default determination, and thus does any resetting of the debt payment, but prior to the insider’s observation
of θt+1 . In the special case where the type is i.i.d. there is no need to maintain the lagged type as a state
variable and there is a single unknown value function V.
Since default occurs if w
e < wid , a firm of type-i taking the type-j allocation defaults for any ε ≤ εdij where
εdij is defined implicitly by
w(b
e j , kj , εdij , θi ) ≡ wid .
(2)
Define Ω as the informed insider’s expectation of the discounted value of total shareholders’ equity given
his true type θi , if he takes the type-j allocation:
Z
Ω(bj , kj , θi ) ≡ βγ
∞
εd
ij
Vi [(1 − δ)kj + π(kj , θi , ε) − bj ]f (dε).
(3)
10 In the U.S., bondholder unanimity is required to change any core provision of public debt outside formal bankruptcy. Most
distressed firms enter the costly Chapter 11 bankruptcy forum in order to restructure public debt.
8
The informed insider maximizes the cum-dividend value of his equity stake, which is equal to
m
m
d+
Ω(b, k, θi ).
c
c+n
(4)
Using the definition of s, the objective function for the insider (4) simplifies to (m/c)Ui (b, d, k, s) where
Z
Ui (bj , dj , kj , sj ) ≡ dj + (1 − sj )βγ
∞
εd
ij
Vi [(1 − δ)kj + π(kj , θi , ε) − bj ]f (dε).
(5)
Conveniently, the multiplicative term m/c has no effect on incentive compatibility constraints. Appendix
A shows the objective function in equation (5) is proportional to the insider’s fractional claim on all future
dividends accounting for dilution.
The function Ui in equation (5) is labeled the insider utility function and is defined on the set of
technologically feasible allocations11
A ≡ {a : d ≥ 0, k ≥ 0, s ≤ 1}.
Given a lagged type j, the option contract (aLj , aHj ) is incentive compatible (IC) if
ICL
:
UL (aLj ) ≥ UL (aHj )
ICH
:
UH (aHj ) ≥ UH (aLj ).
In order to evaluate investors’ response to an option contract, one must derive the fair value of debt. If
w
e < wid for a firm of type-i who took the type-j allocation, lenders demand a revised debt payment, call it
brij < bj , that leaves the firm with the net worth wid . Therefore, we compute brij using
¯ ¯
(1 − δ)kj + π(kj , θi , ε) − brij = wid ⇒ brij = (1 − δ)kj + π(kj , θi , ε) + ¯wid ¯ .
(6)
Equation (6) shows that in default, lenders seize the firm’s capital, operating profits, and an amount |wid |
which represents going-concern value.
Let χ be an indicator function equal to one if bj ≥ 0. The true value of debt issued by type-i under the
type-j allocation is
"
ρ(bj , kj , θi ) ≡ β[χγ + (1 − χ)] bj
Z
Z
∞
f (dε) + (1 − φ)
εd
ij
εd
ij
−∞
#
[(1 − δ)kj + π(kj , θi , ε) +
|wid |]f (dε)
.
(7)
11 An additional constraint is that the firm cannot repurchase more than c − m shares. That is, s ≥ −(c/m − 1). This
constraint never binds for c/m sufficiently large.
9
An incentive compatible option contract is profitable type-by-type if it satisfies the budget constraints
BCi∈{L,H} (w) : di + ki − w ≤ ρ(bi , ki , θi ) + si Ω(bi , ki , θi ).
(8)
The left side of the budget constraint measures the amount of funding provided by investors and the right
side measures the value of securities received in return.
An incentive compatible option contract is profitable in expectation (PIE) given lagged type j if it satisfies
P IE(w, θj ) :
X
p(θi |θj )[ρ(bi , ki , θi ) + si Ω(bi , ki , θi ) − (di + ki − w)] ≥ 0.
i∈{L,H}
Finally, an option contract is interim efficient if it is Pareto optimal (across firm types) within the set
of option contracts that are IC and PIE.
3
Least-Cost Separating Equilibria
Consider a point w on the net worth space and fix two (concave increasing) equity value functions (VL , VH ).
The (least-cost) separating contract (aSL (w), aSH (w)) relative to (VL , VH ) solves
ΓSL (w) ≡ max
a∈A
UL (a)
s.t. BCL (w).
(9)
s.t. BCH (w), ICL .
(10)
and
ΓSH (w) ≡ max
a∈A
UH (a)
The lagged type is irrelevant to the above program.12 It follows that, under separating contracts, allocations
depend on the current reported type but not the lagged type. Therefore, for separating contracts the second
index can be dropped, with
aLL
= aLH ≡ aSL
aHL
= aHH ≡ aSH.
Maskin and Tirole (1992) show that separating allocations are always in the set of PBE.13 Their theorem
is modified here to account for the fact that the equilibrium set in our model varies with net worth. More
importantly, their characterization of the equilibrium set is defined relative to exogenous utility functions.
In our setting, the equilibrium set will ultimately be defined relative to two internally consistent equity value
functions (VL , VH ) which determine the insider utility function per equation (5).
12 By
way of contrast, if we seek to satisfy the PIE constraint, the lagged type will be relevant.
loss of generality we may confine attention to sL ≥ 0. In such cases, Tirole’s weak monotonic profit assumption
is satisfied and the separating contract is equivalent to the low-information-intensity optimum or the Rotschild-Stiglitz-Wilson
contract.
13 Without
10
Proposition 1 (Maskin-Tirole). The set of perfect Bayesian equilibria at net worth w, relative to value
functions (VL , VH ), corresponds to all incentive compatible and profitable in expectation option contracts that
weakly Pareto-dominate the separating contract.
Since the incentive constraint (ICL ) plays an important role, it merits discussion. Rearranging the
equation for ICL , any high type allocation in the feasible set (aH ) must satisfy
S
S
S
[ρ(bSL , kL
, θL ) + Ω(bSL , kL
, θL ) − kL
] − [ρ(bH , kH , θL ) + Ω(bH , kH , θL ) − kH ]
(11)
≥ sH [Ω(bH , kH , θH ) − Ω(bH , kH , θL )] + [ρ(bH , kH , θH ) − ρ(bH , kH , θL )].
The first line of equation (11) measures the efficiency gain to the low type from implementing policies
that maximize his type-specific firm value, as opposed to mimicking the high type. The second line measures
the gain to the low type from masquerading as the high type and issuing mispriced securities. Equation (11)
also illustrates the potency of share repurchases for achieving separation. Specifically, if sH < 0 the low type
is actually harmed by the mispricing of his securities since he is repurchasing shares at inflated prices and
thus diluting the value of his own claim on future dividends.
3.1
Recursive Equilibrium
Proposition 1 characterizes the set of PBE in each period’s signaling game taking the equity value functions
as given.
A sensible definition of equilibrium requires the value functions to be internally consistent.
Conveniently, the equity value functions can be defined recursively by taking expectations over future typecontingent payoffs. We say
d
wj∈{L,H}
∈ <−
eij∈{L,H}×{L,H}
a
:
[wjd , ∞) → A
Vej∈{L,H}
:
[wjd , ∞) → <+
constitute a recursive perfect Bayesian equilibrium (RPBE) if: For each lagged type j and net worth
d
d
eHj (w)) is a PBE relative to (VeL , VeH ); (Endogenous Default) VeL (wL
w ≥ wjd , (e
aLj (w), a
) = VeH (wH
) = 0; and
(Recursivity)
Vej (w) ≡
X
i∈{L,H}
"
p(θi |θj ) dei (w) + [1 − sei (w)] βγ
Z
∞
εd
ii
#
e
e
e
e
Vi [(1 − δ)ki (w) + π(ki (w), θi , ε) − bi (w)]f (dε) . (12)
Proposition 2 constructs an RPBE using a continuum of separating contracts on the net worth space.
11
Proposition 2. Let (aSL , aSH ) be least-cost separating contracts with respect to (VLS , VHS ) where
VjS (w)
≡
X
p(θi |θj )ΓSi (w)
i∈{L,H}
and
w
≡ − max
b,k
ρ(b, k, θL ) + Ω(b, k, θL ) − k.
Then (w, w, aSL , aSH , VLS , VHS ) is a recursive perfect Bayesian equilibrium.
Proof. Proposition 1 states that the separating contract is a PBE. By construction, ViS satisfies recursivity.
By construction, ΓSL (w) = 0. At w, the only allocations satisfying ICL entail dH = 0 and sH = 1. Therefore
ΓSL (w) = 0 ⇒ ΓSH (w) = 0 ⇒ VLS (w) = VHS (w) = 0.¥
Some technical details of the construction are worth noting. First, when we construct an RPBE using
separating contracts, the endogenous default threshold is the point at which the low type is just unable
to satisfy BCL . At that point, the high type cannot possibly separate from the low type and necessarily
receives a payoff of zero. Hence, the default threshold is invariant to firm type in this particular RPBE.
Second, Proposition 1 indicates that for arbitrary w > w, the contract (aSL (w), aSH (w)) is the unique PBE if
and only if it is interim efficient. We return to this point in Section 5, where an RPBE is constructed using
pooling contracts in addition to separating contracts.
3.2
Low Type Policies
To set a baseline, it is useful to characterize policies if outsiders have the same information as the insider.
The superscript F B denotes first-best policies in this case of symmetric information. Under symmetric
B
B
information, optimal debt entails safe debt, e.g. bF
= bF
= 0. With symmetric information, external
L
H
equity dominates defaultable debt since default induces deadweight losses. Saving within the corporate shell
is also dominated under symmetric information since the firm faces catastrophic event risk.
Under symmetric information, optimal investment equates the discounted marginal product of capital
with its price:
1 = βγ[1 − δ + E(π k (kiF B , θi , ε))].
(13)
In order to interpret the first-order conditions for the separating contract, derived in Appendix B, it is
useful to introduce constraint multipliers. Below {µ, λ} denote the (wealth-contingent) multipliers on the
incentive and budget constraints for the high type program determining aSH . Lemma 1 follows from the
first-order conditions determining aSH .
Lemma 1. For the recursive perfect Bayesian equilibrium in Proposition 2, there exists a level of net worth,
12
w,
b such that14
Vi0 (w)
=
1 + p(θH |θi )µ(w)
S
S
Ω(bSH (w), kH
(w), θH ) − Ω(bSH (w), kH
(w), θL )
for w ∈ (w, w)
b
S
S
Ω(bH (w), kH (w), θH )
(14)
= 1 for w ≥ w.
b
Proof. See Appendix C.
Lemma 1 indicates that anticipation of future signaling costs provides a precautionary motive for
accumulating internal resources. The intuition is as follows. If the next realized type is low, a dollar of
internal funds is just worth a dollar, as the firm can finance entirely with equity and on fair terms. However,
if the next realized type is high, a dollar of internal funds is worth more than a dollar since it reduces marginal
signaling costs. This explains why the shadow value of internal resources depends upon the probability of
transitioning to the high type. Note also that the shadow value of internal funds hinges upon a measure of
relative security prices.
The low type’s policies maximize the total value of marketable claims on the firm:
S
(bSL , kL
) ∈ arg max
b,k
ρ(b, k, θL ) + Ω(b, k, θL ) − k.
Proposition 3 follows from the first-order conditions for this problem.
S
S
FB
Proposition 3. For all w ∈ [w, ∞), kL
(w) = kL
> kL
and bSL (w) = bSL ≤ 0 where
Z
βγ
∞
S
S
S
VL0 [(1 − δ)kL
+ π(kL
, θL , ε) − bSL ][1 − δ + π k (kL
, θL , ε)]f (dε) = 1
−∞
· Z ∞
¸
S
0
S
S
S
bL γ
VL [(1 − δ)kL + π(kL , θL , ε) − bL ]f (dε) − 1
= 0.
−∞
Dividends and equity issuance for the low type depend on net worth with
w
<
S
kL
− βbSL ⇒ dSL = 0 and sSL > 0
w
≥
S
S
kL
− βbSL ⇒ dSL = w − (kL
− βbSL ) and sSL = 0.
The intuition for the low type policies is as follows. In order to discourage imitation by the low type, the
separating contract makes the low type as well off as possible, subject to his budget constraint. In a static
signaling model, the low type would receive the symmetric information allocation for his type. In a setting
with repeated signaling and changing types, the low type is given a second-best allocation which accounts
for the precautionary value of internal resources. Consequently, the low type overinvests relative symmetric
S
information and maintains costly financial slack. It is also worth nothing that bSL and kL
are invariant to net
worth. Therefore, the low type satisfies BCL by varying dividends and equity issuance only. This contradicts
14 To
make room for primes we suppress the superscript S from the value function.
13
the pecking-order hypothesis of Myers (1984) which maintains that debt (and only debt) should be used to
achieve budget balance in economies with hidden information.
3.3
High Type Policies
The marginal effect of b on the discounted value of shareholders’ equity for a type-i that has taken the type-j
allocation is
Z
Ωb (bj , kj , θi ) = −βγ
∞
εd
ij
Vi0 [(1 − δ)kj + π(kj , θi , ε) − bj ]f (dε).
(15)
The marginal effect of k on the discounted value of shareholders’ equity for a type-i that has taken the type-j
allocation is
Z
Ωk (bj , kj , θi ) = βγ
∞
εd
ij
Vi0 [(1 − δ)kj + π(kj , θi , ε) − bj ][1 − δ + π k (kj , θi , ε)]f (dε).
(16)
In order to obtain indifference curves over alternative allocations, we compute the total differential of the
insider’s objective function Ui . Indifference curves are defined by
dUi (aj ) = ∆d + (1 − sj )Ωk (bj , kj , θi )∆k + (1 − sj )Ωb (bj , kj , θi )∆b − Ω(bj , kj , θi )∆s = 0.
(17)
From (17), the insider’s willingness to exchange equity for reductions in the face value of debt is
ds
(1 − sj )Ωb (bj , kj , θi )
(aj ; θi ) ≡
.
db
Ω(bj , kj , θi )
(18)
In general, the relative slope of indifference curves in (s, b) space is ambiguous. On one hand, the low type
is more willing to give up equity since he knows his equity is less valuable. On the other hand, the high type
views servicing debt as more costly since he is less likely to default.
From (17), the insider’s willingness to exchange equity for capital is determined by
ds
(1 − sj )Ωk (bj , kj , θi )
(aj ; θi ) ≡
.
dk
Ω(bj , kj , θi )
(19)
Again, there are potentially competing effects at work. On one hand, the low type is more willing to give up
his equity, since his equity is less valuable. On the other hand, the high type has a weakly higher marginal
product of capital. Therefore, the signal content of equity-financed investment is ambiguous absent some
restrictions on primitives.
In Appendix B it is shown that the high type will not pay a dividend if ICL binds. Intuitively, when
the incentive constraint binds, the high type must engage in costly signaling in order to raise external funds
on fair terms. In order to minimize the external funding requirement, the high type pays no dividend if the
incentive constraint binds. The high type’s debt under the separating contract (bSH ) is determined by the
14
following first-order condition:
"Z
βγ
∞
∂εd
S
S
S
S
+ π(kH
, θH , εdHH ) + |w|]
[VH0 ((1 − δ)kH
+ π(kH
, θH , ε) − bSH ) − 1]f (dε) + HH f (εdHH )φ[(1 − δ)kH
∂b
H
εd
HH
·
S
µΩ(bSH , kH
, θL )
=
λ
¯ ¯
¯¸
¸ ·¯
¯ ds S
¯ ¯
¯
¯ (aH ; θL )¯ − ¯ ds (aSH ; θH )¯ .
¯ db
¯ ¯ db
¯
(20)
Equation (20) implies that high type debt trades efficiency against information revelation. The term
VH0 −1 captures a novel deadweight cost of debt service in a dynamic setting with repeated hidden information.
The bondholder values a dollar of debt service at one dollar. However, Lemma 1 indicates that anticipation
of future signaling costs causes shareholders to view the cost of debt service as potentially greater than one.
The next term on the left side of the equation measures marginal default costs. The right side measures
the signal content of debt-for-equity substitutions as measured by the difference between indifference curve
slopes.
S
The first-order condition determining kH
is
"Z
#
∞
1 = βγ
εd
HH
Z
+βγ(1 − φ)
S
S
S
[VH0 ((1 − δ)kH
+ π(kH
, θH , ε) − bSH )][1 − δ + π k (kH
, θH , ε)]f (dε)
εd
HH
−∞
(21)
S
, θH , ε)]f (dε)
[1 − δ + π k (kH
∂εd
S
S
−βγ HH f (εdHH )φ[(1 − δ)kH
+ π(kH
, θH , εdHH ) + |w|]
∂kH
¸
·
¸·
S
ds S
µΩ(bSH , kH
, θL )
ds S
+
(aH ; θH ) −
(aH ; θL ) .
λ
dk
dk
Equation (21) states that high type investment equates marginal benefits with the unit price of capital.
The first term on the right side of the equation measures the marginal benefit to shareholders from an
additional unit of installed capital. Installed capital has a precautionary benefit since VH0 ≥ 1. This effect
encourages overinvestment relative to a setting with symmetric information. The second term measures the
net marginal benefit of investment accruing to bondholders in the event of default. Note that bankruptcy
costs discourage investment, since a portion of the return to capital is lost in the event of default. The
next term measures the marginal benefit of capital accumulation in terms of reducing the region over which
default costs are incurred. The final term measures the signal content of equity-financed investment as
measured by the difference between indifference curve slopes. Again, we see that the separating contract
S
trades efficiency against information revelation. In fact, the pair (kH
, bSH ) equates the ratio of distortion to
information content across real and financial signals.
Proposition 4 spells out some implications of the first-order conditions when the ICL constraint is slack.
15
#
S
S
FB
Proposition 4. For all w ≥ w,
b kH
(w) = kH
> kH
and bSH (w) = bSH ≤ 0 where
Z
∞
βγ
−∞
S
S
S
VH0 [(1 − δ)kH
+ π(kH
, θH , ε) − bSH ][1 − δ + π k (kH
, θH , ε)]f (dε) = 1
· Z ∞
¸
S
S
bSH γ
VH0 [(1 − δ)kH
+ π(kH
, θH , ε) − bSH ]f (dε) − 1
= 0.
−∞
For all w ≥ w,
b dividends and equity issuance for the high type are contingent upon net worth with
S
kH
− βbSH ⇒ dSH = 0 and sSH > 0
w
<
w
S
S
≥ kH
− βbSH ⇒ dSH = w − (kH
− βbSH ) and sSH = 0.
Consider next w < w.
b Characterization of high type policies on this region requires a characterization
of relative indifference curve slopes. Determining the slopes of the indifference curves is complicated by the
fact that unknown value functions, and their slopes, enter into the equations. To gain analytical tractability,
some restrictions are placed on primitives.
Consider first the slope of indifference curves in (b, s) space. On one hand, the low type knows his equity
is less valuable, increasing his willingness to exchange equity for reductions in the face value of debt. On
the other hand, the high type has higher costs of servicing a given debt obligation since he is less likely to
default. Lemma 2 identifies conditions under which the first effect dominates.
Lemma 2. If ε is exponentially distributed and the operating profit function satisfies π εε = 0 and π εθ ≥ 0,
then debt-for-equity substitutions are a positive signal with
¯
¯ ¯
¯
¯ ds
¯ ¯
¯
¯ (a; θL )¯ > ¯ ds (a; θH )¯ ∀ a ∈ A s.t. s < 1.
¯ db
¯ ¯ db
¯
Proof. See Appendix C.
For example, the numerical simulations presented in the next section assume π = θεk α and the conditions
of Lemma 2 are satisfied. In this case, the positive signal content of debt-for-equity substitutions encourages
the high type to take on debt when net worth is sufficiently low such that the incentive constraint binds.
The signal content of capital also reflects competing forces. On one hand, the low type has a strong
incentive to exchange equity for capital since he knows his equity is less valuable. On the other hand, the
high type can use a marginal unit of capital more productively. Lemma 3 presents sufficient conditions such
that the first effect dominates.
Lemma 3. If ε is exponentially distributed and the operating profit function satisfies π εε = 0, π kε = 0 and
π kθ
π εθ
≤
,
1 − δ + πk
πε
16
then equity financed investment is a negative signal with
ds
ds
(a; θL ) >
(a; θH ) ∀ a ∈ A s.t. s < 1.
dk
dk
Proof. See Appendix C.
If asymmetric information concerns only the value of assets in place, with π = θε + k α , for example, the
conditions of Lemma 3 are satisfied. Effectively, this particular functional form ensures that the marginal
product of new capital is independent of type, so signal content hinges upon relative equity valuations. Since
the low type has less valuable equity, he is more willing to exchange equity for capital.
Taken together, Lemmas 2 and 3 provide the following insights. First, when the conditions of Lemma
2 are satisfied, equation (20) tells us that the high type will have more debt than the second-best level if
the incentive constraint binds. By way of contrast, even if the conditions of Lemma 3 are satisfied, the net
effect of asymmetric information on capital accumulation for the high type remains ambiguous due to the
competing factors entering into equation (21). Settling this particular question therefore requires simulating
the full model. This task is taken up in the next section.
4
Model Estimation and Simulation
A unique feature of the model is that it endogenizes all relevant controls of the firm: debt and equity issuance,
investment, dividends, repurchases, and default. The realism of the model allows us to take it directly to
the data. In this section, we begin by fitting the model to real-world moments. We then determine whether
the calibrated model does a good job in matching stylized facts. Additionally, the calibration exercise itself
serves as a check on the model. In particular, the model would be questionable if it required implausible
parameter values in order to fit the data.
4.1
Calibration and Estimation
Table 1 provides detail on model parameters. In order to reduce computation time, structural parameters
are taken from existing literature when feasible. Cooley, Marimon and Quadrini (2004) set the depreciation
rate δ = 0.0579. Gomes (2001) assumes δ = 0.145. We split the difference and set δ = 0.10. Following
Carlstrom and Fuerst (1997) we set the real interest rate r = 0.04 and the probability of a catastrophic
shock 1 − γ = 0.05. The operating profit function is π = θεk α with ε exponentially distributed with mean
and standard deviation of 1/2. Low type productivity (θL ) is normalized to 0.4. The choice of the two
preceding parameters is arbitrary, but the assumed values do affect the necessary grid space and speed of
numerical calculations. To reduce the number of unknown parameters, the assumed transition matrix for θ
is symmetric with p(θi |θi ) ≡ p. In addition to the persistence parameter p, we must estimate: the curvature
of the profit function (α); high type productivity (θH ); and bankruptcy costs (φ).
17
We use a version of simulated method of moments (SMM) where parameters are chosen to minimize
the distance between data moments and moments generated by the simulated model. The procedure used
to solve the model numerically is described in Appendix D. In principal, one could estimate the remaining
unknown parameters by applying SMM to the full model with asymmetric information. However, this is
not feasible given the run-time of the full model.15 Instead, we split the estimation into two stages. In
the first-stage estimation, we use SMM to estimate the parameters of the profit process (p, α, θH ) based on
moments from a sample of firms that have bond ratings. Fazzari, Hubbard and Petersen (1988), Whited
(1992), and Cooper and Ejarque (2001) use bond ratings as a proxy for “lack of financial constraint.” We use
bond ratings as an indicator for lack of asymmetric information. In the absence of asymmetric information,
model run-time is much shorter since here the simulated firm always implements first-best investment. The
first-stage estimation utilizes 2000 simulated panels, each consisting of 2000 firms for 30 periods. This size
panel is standard in empirical corporate finance.
In the second-stage estimation, we apply SMM to the full model with asymmetric information,
parameterized with the first-stage estimates of (p, α, θH ), to identify the bankruptcy cost (φ). In particular,
the full model is calibrated so as to fit a sample of relatively young post-IPO firms who are likely to encounter
asymmetric information. The second-stage estimates utilize 2000 simulated panels, each consisting of 2000
firms for 10 periods. Each of the simulated firms starts as a high type and with zero net worth. The intent
is to mirror the data generating process in the study of Baker and Wurgler (2002), who follow a panel of
firms ten years post-IPO.
In each of the two stages, the goal is to estimate a vector of structural parameters, with true value
denoted ξ ∗ . Let X be the real-world data and consider a function of the observed sample FT (X), where T
is the sample time period. The statistic FT (X) could represent a collection of sample means or even a more
complicated estimator, such as regression slope coefficients. Assume that as the sample size T increases,
FT (X) converges in probability to limit M (ξ ∗ ). Next, let xn be the corresponding model generated data of
length T , where n indexes the simulation round. Let
N
1 X
mN (ξ) ≡
FT (xn ; ξ)
N n=1
(22)
represent an estimate of M (ξ) based on N independent model simulations evaluated at an arbitrary vector
ξ. Let
GN (X, ξ) ≡ mN (ξ) − FT (X),
(23)
denote the difference between the estimated statistic F and its observed empirical value. Under appropriate
regularity conditions, it can be shown that as the sample size (T ) and the number of simulations (N ) both
15 Due
to the incentive constraint, the feasible set of the high type is much harder to compute and requires a very fine grid.
18
go to infinity, the minimum distance estimator
b
ξ ≡ arg min GN (X, ξ)0 WT GN (X, ξ)
ξ
(24)
is a consistent estimator for ξ ∗ . The matrix WT in the above expression is a positive definite matrix that
converges in probability to a deterministic positive definite matrix W .
Let Ψ denote the asymptotic variance-covariance matrix of FT (X). The efficient weighting matrix is
W = Ψ−1 . The corresponding estimator b
ξ is consistent and asymptotically normal with variance-covariance
matrix (J 0 Ψ−1 J)−1 , where J ≡ ∇ξ M (ξ).
We initially use the identity matrix for the weighting matrix to get an initial estimate of ξ ∗ , which is
denoted e
ξ. In the next estimation step, the weighting matrix is set equal to the inverse of the estimated
variance-covariance matrix from the first step:
N
X
b≡ 1
[mN (e
ξ) − FT (xn ; e
Ψ
ξ)][mN (e
ξ) − FT (xn ; e
ξ)]0 .
N n=1
(25)
In order to estimate (p, α, θH ), the symmetric information model is matched to moments from bond rated
firms in the sample used by Riddick and Whited (2007).16 The moments to be matched are: the mean of
Tobin’s Q; the standard deviation of Q; the serial correlation of Q; and the slope from a regression of the
investment rate on Q. Values of Q coming from real-world data are likely to contain measurement error.
However, Q can be measured without error in the simulated data. To induce symmetry between the two
data generating processes, we multiply model-generated Q values by a log-normal scalar with mean one and
variance σ. Thus, σ becomes another free variable to be estimated.
As shown in Table 1, the estimated parameter values from the first-stage estimation are: α = 0.7488;
p = 0.9325; θH = 0.7759; and σ = 0.7692. The estimated elasticity of profit with respect to capital is in line
with the estimate of α = 0.69 obtained by Cooper and Ejarque (2001). The high estimated value of p is
consistent with a high degree of serial correlation in profits, with the public shock (ε) adding noise. The high
type productivity factor is nearly twice as large as that of the low type. Table 2 compares data moments
with moments from the simulated model. The model fits well the first two moments of Q. However, the
calibrated model undershoots serial correlation in observed Q. A potential explanation for the divergence
is serial correlation in real-world Q measurement error. Finally, the model overshoots the investment-Q
reponsiveness. Again, measurement error in Q offers a potential explanation.
Baker and Wurgler (2002) track leverage ratios from a comprehensive sample of U.S. firms post-IPO.
This sample is relevant in the context of our calibration since relatively young firms are more likely to face
problems associated with asymmetric information. As reported by Baker and Wurgler, the average market
leverage ratio ten years after the IPO is 0.4373. We calibrate the model with asymmetric information in
16 We
thank Toni Whited for providing us with these moments.
19
order to match this value. The minimum distance estimate is φ = 21%, producing an average leverage
ratio of 0.4524. By way of contrast, Weiss (1990) directly estimates that average legal costs in bankruptcy
amount to only 2.8% of the book value of assets. Andrade and Kaplan (1998) estimate that indirect costs
of bankruptcy ranging from 10% to 20% of total firm value. Using SMM, Hennessy and Whited (2007)
estimate bankruptcy costs of 10% for their full sample and 15% for small firms. Although relatively high,
the estimated bankruptcy cost is consistent with existing estimates.
4.2
Simulation Output
This subsection discusses the behavior of the calibrated model. To illustrate concavity of the equity value
function, or pseudo-risk-aversion, we begin by plotting the firm’s enterprise value. Enterprise value (ei )
measures the difference between equity value and net worth, ei (w) ≡ Vi (w) − w. The slope of the enterprise
value function is a direct measure of the precautionary value internal funds, since e0i = Vi0 − 1. Alternatively,
one can think of e0i as measuring the gain that shareholders would capture if they could directly inject cash
into the firm just before the next realized type is observed. Recalling that η denotes the multiplier on the
dividend nonnegativity constraint in the high type program, we can express the enterprise value as
Z
ei (w)
w
= |w| +
w
=
[Vi0 (ω) − 1]dω
Z
(26)
w
|w| + p(θH |θi )
η(ω)dω
∀ w > w.
w
Thus, enterprise value is equal to going-concern value plus the cumulative precautionary value of internal
resources. Figure 2 plots the two enterprise value functions. Consistent with concavity of both equity value
functions (VL , VH ), the slopes of both enterprise value functions are positive and decreasing in net worth.
The enterprise value has a slope of zero for high levels of net worth where the incentive constraint is slack.
Consistent with equation (26), the slope of the enterprise value function is particularly steep for a firm with
a high lagged type. Intuitively, internal funds have high precautionary value when there is a high probability
of transitioning to a high type and incurring signaling costs.
Figure 3 plots the capital allocations of each type relative to first-best. The low type invests above firstbest regardless of realized net worth. The overinvestment of the low type reflects the fact that informational
asymmetries create a precautionary motive for capital accumulation. The high type generally underinvests
relative to first-best, with investment increasing in net worth. Thus, the model is consistent with the findings
of Blanchard, Lopez-de-Silanes and Shleifer (1994) and Rauh (2006) who document that exogenous increases
in internal resources induce increases in capital expenditures.
It is worth noting that Fazzari, Hubbard and Petersen(1988) also predict that exogenous cash windfalls
will induce increases in capital expenditures. Their graphical model assumes a cost of capital schedule that
is increasing in external funds, but invariant to investment. In that setting, a cash windfall moves the firm
to a lower cost of capital and boosts investment. The causation in our model is different. When net worth
20
is low, firms with positive information rely upon risky debt financing to signal positive information. As
shown in equation (21), the possibility of costly default reduces the expected return to investment. Thus,
underinvestment emerges. Next consider the effect of a cash windfall on the high type. A marginal dollar can
be used to fund new investment without violating incentive compatibility. Further, the increase in capital
would crowd-in more investment, since it lowers default risk at the margin.
Figure 4 (Panel B) plots the wealth-contingent financing policies of the low type. Consistent with
Proposition 3, the low type uses dividends and equity issuance as the sole means of achieving budgetbalance, while retaining a wealth-invariant level of savings. When net worth is low, the low type sets the
dividend to zero and issues a large amount of equity. Equity issuance for the low type declines monotonically
with net worth, while the dividend increases with net worth.
Figure 4 (Panel A) plots the wealth-contingent financing policies of the high type. If net worth is
sufficiently high, the high type initiates dividends and share repurchases. Thus, the model is consistent with
the positive relationship between corporate distributions and internal resources documented by DeAngelo,
DeAngelo and Stulz (2006). When the high type issues equity, it generally conducts a joint offering which
combines equity and debt. In contrast, the low type issues equity without any debt. This is consistent with
existing empirical studies. Asquith and Mullins (1986) document negative abnormal returns in a sample of
pure common stock offerings. Masulis and Korwar (1986) find that seasoned equity offerings are associated
with negative price changes on average. However, the announcement return is positively related to leverage
changes.
The debt level of the high type is a nonmonotone function of net worth. When net worth is low, marginal
increases in net worth induce increases in debt. As discussed above, when net worth is low, marginal increases
in net worth induce more investment. Since this reduces default costs, marginal increases in debt are optimal.
Returning to Figure 3, we see that when net worth is sufficiently high, the high type no longer increases
investment in response to cash windfalls. Rather, cash windfalls are used as a substitute for debt finance.
Figure 5 shows that the leverage ratio of the high type is decreasing in net worth. In untabulated regressions
using a panel of simulated firms, the market leverage ratio was found to be mean-reverting and decreasing
in lagged profitability, consistent with the empirical findings of Fama and French (2002).
The model provides a laboratory for analyzing the cross-sectional determinants of abnormal returns
associated with investment and financing announcements. In the simulated model data, one can perfectly
isolate the policy announcement effect. The stock price just prior to the policy announcement at time t
is equal to Vθt−1 (wt )/c. The stock price immediately following the policy announcement is Γθt (wt )/c. The
pure abnormal return (AR) associated with the announced policy is the ex post price less the ex ante price
normalized by the ex ante price
ARt ≡
Γθt (wt ) − Vθt−1 (wt )
.
Vθt−1 (wt )
(27)
Table 3 analyzes the determinants of model-implied abnormal returns, treating the simulated ARt as a
dependent variable. Such regressions are common to the event study literature. A common theme running
21
through the regressions is that the announcement of high investment should induce high abnormal returns.
This is consistent with the empirical evidence presented by McConnell and Muscarella (1985) who document
raw equity returns of 1.21% following announced increases in capital budgets and -1.52% for decreases. In
the second regression, we see a leverage ratio is also a positive signal. This is consistent with the findings of
Masulis (1983) who documents a 13.97% primary announcement return in debt for equity exchanges and a
-9.91% return in equity for debt exchanges. In the third regression, we see that capital expenditures are the
most important driver of abnormal returns.
5
Pooling Equilibria
Prior sections confined attention to separating allocations, which are always in the set of PBE relative to
fixed equity value functions. From a continuum of such separating contracts, a recursive PBE can always be
constructed. This section identifies points on the net worth space at which it is possible to sustain pooling.
Conveniently, the theorem of Maskin and Tirole (1992) indicates a PIE pooling contract is a PBE only if
it weakly Pareto-dominates the separating allocations. Using this Pareto criterion to identify pooling PBE,
we can update the equity value functions to reflect changes in PBE, and vice versa.
d
d
When attention is confined to separating equilibria, the default threshold is given by wL
= wH
=w, as
defined in Proposition 2. Anticipating, the selection of Pareto-dominant pooling allocations increases the
firm’s continuation region. The intuition is simple. At the net worth level w, both types receive zero under
the separating contracts because a low type, revealed as such, cannot satisfy his budget constraint. At this
level of net worth, both types would be better off pooling at some PIE allocation. Intuitively, when net worth
is extremely low, the high type must take on extremely high levels of debt in order to achieve separation.
At such low levels of net worth, the high type is better off pooling at some allocation with zero debt, despite
the fact that his equity is underpriced.
We consider PIE pooling allocations that maximize the expected payoff to the insider just prior to his
private observation of the current type. As shown below, such allocations can also be viewed as “focal” since
they also maximize the firm’s ability to meet outstanding debt obligations. Recall that separating allocations
depend on the current type but not the lagged type. By way of contrast, the pooling allocation depends on
the lagged type but not the current type. For the pooling contract we may write
aLL
= aHL ≡ aP
L
aLH
= aHH ≡ aP
H.
Let aP
j (w) denote the pooling allocation at net worth w, given the lagged type is j. This pooling allocation
solves
aP
j (w) ∈ arg max
a∈A
d + (1 − s)
X
p(θi |θj )Ω(b, k, θi ) s.t. P IE(w, θj ).
i∈{L,H}
22
(28)
Substituting the PIE constraint into the maximand, it follows that the pooling allocation maximizes the
total expected value of marketable claims on the firm:
X
P
(bP
j , kj ) ∈ arg max
b,k
p(θi |θj )[ρ(b, k, θi ) + Ω(b, k, θi )] − k.
(29)
i∈{L,H}
P
It follows that (bP
j , kj ) do not depend on w. Rather, as w increases, the PIE constraint is satisfied by
reducing equity flotations or increasing dividends. However, it is interesting to note that the lagged type
P
does affect (bP
j , kj ) since the lagged type determines transition probabilities.
In the conjectured recursive PBE, the pooling allocations are implemented on some right neighborhood
of the default threshold. Default occurs when the firm is just unable to satisfy the PIE constraint. The
resulting default threshold for a firm with lagged type j is
X
wj ≡ − arg max
b,k
p(θi |θj )[ρ(b, k, θi ) + Ω(b, k, θi )] − k.
(30)
i∈{L,H}
The equation for the endogenous default threshold (30) has two interesting features. First, the default
threshold now depends on the firm’s lagged type. This is to be expected. Here default occurs when the PIE
constraint cannot be satisfied. The lagged type affects the PIE constraint since the lagged type determines
transition probabilities. For example, with serial correlation in types, wH < wL . Second, comparison of
equation (30) with the endogenous default threshold w given in Proposition 2 indicates that for either lagged
type j, wj <w. This reflects the better terms offered by the PIE constraint relative to the constraint BCL .17
The first-order conditions for the pooling contract given lagged type j are:
X
Z
∞
p(θi |θj )βγ
−∞
i∈{L,H}
"
bP
j
γ
P
Vi0 [(1 − δ)kjP + π(kjP , θi , ε) − bP
j ][1 − δ + π k (kj , θ i , ε)]f (dε)
X
Z
∞
p(θi |θj )
i
−∞
=
1
(31)
#
Vi0 [(1
−
δ)kjP
+
π(kjP , θi , ε)
−
bP
j ]f (dε)
−1
bP
j
= 0
≤ 0.
Note that no debt is issued under the pooling contract. This is because debt carries with it default and
deadweight costs stemming from the fact that the shadow value of internal funds potentially exceeds one. In
a pooling equilibrium there is no point to incurring such costs since financial structure does not signal private
information. The capital stock in the pooling allocation essentially splits the difference between the optimal
type-specific capital stocks. Once capital and debt have been computed, dividends and equity flotations are
pinned down by the PIE constraint.
The next step in the construction is to choose the pooling allocation and payoffs only if they Pareto17 Since
Pareto-dominant pooling equilibria are selected, the value functions also increase endogenously. Thus, the inequality
holds a fortiori.
23
dominate the separating allocations. For this purpose, let ΓP
ij denote the payoff to the insider of type-i under
the pooling contract given the lagged type was j:
P
ΓP
ij (w) ≡ Ui (aj (w)).
Next, let Γ∗ij (w) denote the PBE payoff to the insider of type-i given lagged type-j and let a∗ij (w) denote the
corresponding allocation. Since pooling can only be supported as a PBE if it Pareto-dominates the payoffs
to the separating contract we have
ΓP
ij (w)
∗
P
≥ ΓSi (w) ∀i ⇒ Γ∗ij (w) = ΓP
ij (w) and aij (w) = aj (w)
∀i
(32)
else
Γ∗ij (w) =
ΓSi (w) and a∗ij (w) = aSi (w)
∀i.
The next step in the construction is to define the equity value functions recursively:
Vj∗ (w) ≡
X
p(θi |θj )Γ∗ij (w).
(33)
i∈{L,H}
It follows that Vj∗ inherits the properties of the Γ∗ functions. For example, on the pooling region the Γ∗
functions are in fact linear in net worth. On the region where separation occurs, Γ∗Hj = ΓSHj and the value
function has a slope exceeding one if the incentive constraint binds.
Proposition 5 provides a formal characterization of an RPBE that features regions of pooling and
separation.
Proposition 5. For each lagged type there exists an interval [wi , wiP ] such that the pooling allocation is in
the set of PBE if and only if w ∈[wi , wiP ].
Proof. Choose ² arbitrarily small and let w ≡ wi + ². The separating payoffs at this point are zero since the
feasible set is empty. However, if the types pool, they achieve strictly positive payoffs at some s < 1. From
Proposition 1 it follows that pooling is here in the set of PBE. Further ∃ w
b s.t. ICL is slack ∀w ≥ w.
b Thus,
for either lagged type j, ΓSH (w) > ΓP
b From Proposition 1 it follows that the pooling allocation
Hj (w) ∀w ≥ w.
is not a PBE on this region.¥
The intuition for Proposition 5 is simple. If and only if net worth is sufficiently low, the costs of signaling
swamp the value to the high type of separating and receiving fair value for his securities. In such cases, the
pooling allocation Pareto-dominates the separating allocation.
24
6
Conclusions
This paper takes a first step in constructing dynamic structural models of corporate financing when insiders
have private information. We show that anticipation of future signaling costs converts a risk-neutral insider
into a pseudo-risk-averse insider. The implied precautionary value of internal funds discourages debt and
encourages capital accumulation relative to what one obtains in a static setting. In the least-cost separating
equilibrium, firms signal positive information with high leverage and underinvest relative to first-best. Firms
with negative information use only equity finance and overinvest relative for first-best. Finally, we argue that
the nature of equilibrium should be contingent upon net worth. In particular, if net worth is sufficiently low,
the costs of separation are extremely high and firms should find their way to a Pareto dominating pooling
equilibrium.
The most obvious direction to take this line of research is to analyze optimal security design. In this
paper, we took the set of securities as given, and derived equilibria given this set. We anticipate that the
securities emerging as optimal under repeated hidden information will be similar to those that are optimal
under repeated moral hazard. Whether the problem is one of hidden information or hidden action, low
reported values of operating profits must be punished. In our model, low reported profits are punished with
transfers of ownership in default. A more general model, with long-term debt, would allow for intermediate
punishments in the form of higher interest rates or other changes in credit terms.
25
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28
Appendix A: The Insider’s Objective Function
In an arbitrary period t, the privately informed insider has been assumed to maximize his payoff as if he
were to hold for a single period
U ≡ dt + (1 − st )βγEt [Vt+1 ] = dt + βγ
ct
ct+1
Et [Vt+1 ].
Using the law of iterated expectations
·
Et [Vt+1 ]
=
=
Et [Vt+2 ]
=
ct+1
Et+1 [Vt+2 ]
Et dt+1 + βγ
ct+2
·
¸
ct+1
Et dt+1 + βγ
Vt+2 .
ct+2
·
¸
ct+2
Et dt+2 + βγ
Vt+3 .
ct+3
¸
Substitution of these terms into the function U yields
·
µ
¶
¶
¶¸
µ
µ
dt+1
dt+2
Vt+3
2
3
U = dt + ct Et (βγ)
+ (βγ)
+ (βγ)
.
ct+1
ct+2
ct+3
Iterating in this fashion, one obtains
"∞
µ
¶#
X
U
dt+τ
τ
(βγ)
.
= Et
ct
ct+τ
τ =0
Thus, maximizing U is equivalent to maximizing the expected discounted value of the future dividend stream
coming from one share of stock.
Appendix B: First-Order Conditions for Separating Contracts
S
The first-order conditions for interior (bSL , kL
) are
S
S
ρb (bSL , kL
, θL ) + Ωb (bSL , kL
, θL )
=
0
S
S
ρk (bSL , kL
, θL ) + Ωk (bSL , kL
, θL )
=
1.
There is a kink in the debt value function at b = 0, and thus it is possible to have bSL = 0. We know
bSL > 0 cannot be optimal since the right derivative ρb + Ωb < 0 for all bL ≥ 0.
The Lagrangian for aSH is
LH (w) ≡
d + (1 − s)Ω(b, k, θH ) + λ[ρ(b, k, θH ) + sΩ(b, k, θH ) + w − k − d]
+µ[ΓSL (w) − d − (1 − s)Ω(b, k, θL )] + ηd + ψ(1 − s).
29
The first-order conditions for dSH and sSH are
1−λ−µ+η =0
S
S
(λ − 1)Ω(bSH , kH
, θH ) + µΩ(bSH , kH
, θL ) = ψ.
Rearranging these equations one obtains
S
S
S
ηΩ(bSH , kH
, θH ) − µ[Ω(bSH , kH
, θH ) − Ω(bSH , kH
, θL )] = ψ.
It is straightforward to establish ψ = 0 on the continuation region. Suppose to the contrary ψ > 0 ⇒ η > 0.
Thus, the high type gets a payoff of zero, which contradicts being on the continuation region. Since ψ = 0
we obtain
·
η=µ
¸
S
S
, θH ) − Ω(bSH , kH
, θL )
Ω(bSH , kH
.
S ,θ )
Ω(bSH , kH
H
Thus µ > 0 ⇒ dSH = 0.
The first-order condition for bSH is
S
S
S
S
(1 − sSH )Ωb (bSH , kH
, θH ) + λ[ρb (bSH , kH
, θH ) + sΩb (bSH , kH
, θH )] = µ(1 − sSH )Ωb (bSH , kH
, θL ).
Rearranging terms one obtains
·
S
S
ρb (bSH , kH
, θH ) + Ωb (bSH , kH
, θH ) =
S
µ(1 − sSH )Ω(bSH , kH
, θL )
λ
¸·
¸
S
S
Ωb (bSH , kH
, θL ) Ωb (bSH , kH
, θH )
−
.
S ,θ )
S ,θ )
Ω(bSH , kH
Ω(bSH , kH
L
H
S
The first-order condition for kH
is
S
S
S
S
(1 − sSH )Ωk (bSH , kH
, θH ) + λ[ρk (bSH , kH
, θH ) + sΩk (bSH , kH
, θH ) − 1] = µ(1 − sSH )Ωk (bSH , kH
, θL ).
Rearranging terms one obtains
·
S
, θH )
ρk (bSH , kH
+
S
, θH )
Ωk (bSH , kH
S
µ(1 − sSH )Ω(bSH , kH
, θL )
−1=
λ
¸·
¸
S
S
Ωk (bSH , kH
, θL ) Ωk (bSH , kH
, θH )
−
.
S ,θ )
S ,θ )
Ω(bSH , kH
Ω(bSH , kH
L
H
Appendix C: Proofs of Lemmas
Lemma 1. For the recursive perfect Bayesian equilibrium in Proposition 2, there exists a level of net worth,
30
w,
b such that
Vi0 (w)
·
¸
S
S
Ω(bSH (w), kH
(w), θH ) − Ω(bSH (w), kH
(w), θL )
= 1 + p(θH |θi )µ(w)
for w ∈ (w, w)
b
S (w), θ )
Ω(bSH (w), kH
H
= 1 for w ≥ w.
b
(34)
Proof. First note ICL must bind at w since the high type gets a zero payoff here but would achieve a
strictly positive payoff if he needed to satisfy only BCH . Thus, ICL binds for sufficiently low net worth.
Next consider high levels of net worth and maximize the utility of the high type subject only to BCH . For
w sufficiently high the solution to this relaxed program entails sH = 0 and bH ≤ 0. Thus, the high type is
not floating any securities and the low type cannot gain from mimicry since there is no gain from securities
mispricing.
Next, we derive Vi0 for net worth levels such that ICL binds. We know
Vi0 (w)
0
0
= p(θH |θi )ΓH (w) + p(θL |θi )ΓL (w)
0
= p(θH |θi )ΓH (w) + p(θL |θi )
h 0
i
= 1 + p(θH |θi ) ΓH (w) − 1
= 1 + p(θH |θi )[L0H (w) − 1]
= 1 + p(θH |θi )[λ(w) + µ(w) − 1]
= 1 + p(θH |θi )η(w).
The second line follows from the linearity of ΓSL . The fourth line follows from the Envelope Theorem. The
rest follows from the first-order conditions pinning down η.¥
Lemma 2. If ε is exponentially distributed and the operating profit function satisfies π εε = 0 and π εθ ≥ 0,
then debt-for-equity substitutions are a positive signal with
¯ ¯
¯
¯
¯ ds
¯ ¯
¯
¯ (a; θL )¯ > ¯ ds (a; θH )¯ ∀ a ∈ A s.t. s < 1.
¯ db
¯ ¯ db
¯
Proof. Note that
dVi
= Vi0 (w)π ε .
dε
Using this equation and the hypothesis π εε = 0 we obtain
¯
¯
¯
¯ ds
¯ (aj ; θi )¯
¯
¯ db
=
R∞
(1 − sj ) εd (dVi /dε)f (dε)
R ∞ij
π ε εd (Vi )f (dε)
=
(1 − sj )E(ε)
.
πε
ij
31
The second line uses integration-by-parts and the hypothesis of ε having the exponential distribution.¥
Lemma 3. If ε is exponentially distributed and the operating profit function satisfies π εε = 0, π kε = 0 and
π kθ
π εθ
≤
,
1 − δ + πk
πε
then equity financed investment is a negative signal with
ds
ds
(a; θL ) >
(a; θH ) ∀ a ∈ A s.t. s < 1.
dk
dk
Proof. Following the same steps as in Lemma 2 we obtain
·
¸
ds
(1 − δ + π k )
(aj ; θi ) =
(1 − sj )E(ε).
dk
πε
The stated hypothesis ensures the term in squared brackets is strictly decreasing in θ.¥
Appendix D: Details of the Numerical Algorithm
The solution procedure is based on value function iteration. The individual steps are as follows. The
idiosyncratic shock ε is implemented by discretizing its domain using N possible values. Each maximization
is implemented by discretizing the domain of the decision variables.
1. Guess default threshold w.
2. Guess the end-of-period equity value functions (VL , VH ) which are vectors on the net worth space.
3. For each point in the net worth grid, find the allocation aL that maximizes the objective function of
the low type subject to its budget constraint. Since the dividend is not unique, pick the allocation in the
optimal set that minimizes the dividend.
4. For each point in the net worth grid, find the allocation aH that maximizes the high type’s objective
subject to the budget and nonmimicry constraints.
0
5. Using the solutions from steps 3 and 4, compute new value functions Vj using the recursive equation
"
0
Vj = p(θH |θj ) dH + βγ(1 − sH )
"
+ p(θL |θj ) dL + βγ(1 − sL )
N
X
#
f (εn )VH [(1 − δ)kH +
n=1
N
X
α
θH εn kH
− bH ]
#
α
f (εn )VL [(1 − δ)kL + θL εn kL
− bL ] .
n=1
0
6. The functions Vj from the previous step are the new guesses for Vj . The procedure is then restarted
from step 2 until convergence.
7. Check the endogenous default condition Vj (w) = 0. If the condition is not satisfied, update the initial
guess w and restart the procedure from step 1 until convergence.
32
Figure 1: Timeline
t+1
t
6
•
•
•
•
•
•
time
-
6
θ t−1 and wt are common knowledge
Insider privately observes θt
Insider offers option contract
Investor accepts/rejects
If accept, insider chooses allocations
Catastrophic risk exposure
•
•
•
•
•
•
33
εt is realized
θt is inferred by outsiders
w
et+1 = (1 − δ)kt + π(kt , θ t , εt ) − bt
Lender declares default/not
wt+1 = max{wθdt , w
et+1 }
Sequence repeats
Figure 2: Enterprise value of the firm
The enterprise value of the firm Vi − w is plotted as a function of the realized net worth, w. The productivity shock ε is
exponentially distributed and we use parameters reported in Table 1 in calculations.
2.5
Enterprise Value
2
1.5
H
L
1
−2
0
2
4
w
34
6
8
10
Figure 3: Equilibrium capital allocations
Equilibrium capital allocations, kiS , scaled by the first-best allocations, kiF B , are plotted as a function of the realized net worth,
w, for both high and low values of θ. The productivity shock ε is exponentially distributed and we use parameters reported in
Table 1 in calculations.
1.2
H
L
1.1
1
0.9
k*/kFB
0.8
0.7
0.6
0.5
0.4
0.3
0.2
−2
0
2
4
w
35
6
8
10
Figure 4: Equilibrium financing policies
S
Equilibrium financing policies: debt, ρi , and equity, sS
i Ω, as well as equilibrium dividend policy, di , are plotted as functions of
the realized net worth, w. Panel A presents the case of high value of θ, while Panel B presents the case of low value of θ. The
productivity shock ε is exponentially distributed and we use parameters reported in Table 1 in calculations.
Panel A: High value of θ
5
Debt
Total Issued Equity
Dividends
4
3
2
Value
1
0
−1
−2
−3
−4
−5
−2
0
2
4
w
6
8
10
6
8
10
Panel B: Low value of θ
10
9
Debt
Total Issued Equity
Dividends
8
7
Value
6
5
4
3
2
1
0
−2
0
2
4
w
36
Figure 5: Book and market leverage
High type’s book leverage,
ρ
kiS
, and market leverage,
ρ
ρ+Ω
are plotted as functions of the realized net worth, w. The public
shock ε is exponentially distributed and we use parameters reported in Table 1 in calculations.
1.8
Market Leverage
Book Leverage
1.6
1.4
Leverage Ratio
1.2
1
0.8
0.6
0.4
0.2
0
−2
0
2
4
w
37
6
8
10
Table 1: Parameter Values
This table reports parameter values used to solve and simulate the model. Panel A reports exogenously chosen parameters and
Panel B reports SMM-estimated parameters.
Notation
Parameter Value
Definition
Panel A: Exogenously Chosen Parameters
r
1−γ
δ
ε
θL
0.0400
0.0500
0.1000
0.5000
0.4000
α
θH
p(θi |θ i )
φ
σ
0.7488
0.7759
0.9325
0.2091
0.7692
Risk-Free Rate
Catastrophic Event Probability
Capital Depreciation Rate
Mean of Public Shock
Low Type Productivity
Panel B: Estimated Parameters
Capital Elasticity of Profits
High Type Productivity
Type Persistence
Proportional Bankruptcy Costs
Standard Deviation of the Measurement Error in Tobin’s Q
Table 2: Model and Data Moments
This table reports moments implied by the model. The first four moments are obtained from estimating the full information
version of the model. The fifth moment is obtained from estimating the asymmetric information version of the model. In both
cases we use a two-step SMM procedure where a panel of 2000 firms is simulated 2000 times. The full information model is
simulated for 200 time periods and only the last 30 observations are kept. In the asymmetric information case, all firms start
with positive information (θ = θH ) and zero net worth and are simulated for 10 time periods all of which are kept. In both
cases we use exponentially distributed public shocks.
Moment
Data
Model
Average Tobin’s Q
2.6780
2.7241
Standard deviation of Tobin0 s Q
3.6990
3.6774
Serial correlation of Tobin0 s Q
0.6450
0.2098
Slope coefficient from Investment-Q regression
0.0260
0.2021
0.4373
0.4524
Market leverage (IPO + 10)
ρ(bi ,ki ,θ i )
ρ(bi ,ki ,θ i )+Ω(bi ,ki ,θ i )
38
Table 3: Announcement Effect Regressions
This table reports results of several regressions on the simulated data with the abnormal return on the announcement day,
ARt = (Γθt (wt ) + Vθt−1 (wt ))/Vθt−1 (wt ), as the dependent variable. The investment rate is defined as kt+1 /kt − (1 − δ).
Notation a± means conditioning on the positive(negative) values of a. The simulated panel of firms contains 2,000 firms over
30 time periods, where the initial period has been dropped for each firm. The public shock ε is exponentially distributed and
we use parameters reported in Table 1 in simulations.
Investment Rate
0.0793
( 0.0202 )
ρt
kt
-
0.2078
( 0.0518 )
0.0566
( 0.0211 )
0.0433
( 0.0352 )
-
39