Download Ineffi cient Investment Waves - The University of Chicago Booth

Ine¢ cient Investment Waves
Zhiguo He
University of Chicago and NBER
Péter Kondor
London School of Economics and CEPR
August 2015
forthcoming in Econometrica
Abstract
We show that …rms’ individually optimal liquidity management results in socially ine¢ cient boom-and-bust patterns. Financially constrained …rms decide on the level of their liquid
resources facing cash-‡ow shocks and time-varying investment opportunities. Firms’ liquidity
management decisions generate simultaneous waves in aggregate cash holdings and investment,
even if technology remains constant. These investment waves are not constrained e¢ cient in general, because the social and private value of liquidity di¤ers. The resulting pecuniary externality
a¤ects incentives di¤erentially depending on the state of the economy, and often overinvestment
occurs during booms and underinvestment occurs during recessions. In general, policies intended
to mitigate underinvestment raise prices during recessions, making overinvestment during booms
worse. However, a well-designed price-support policy will increase welfare in both booms and
recessions.
Key Words: Pecuniary externality, overinvestment and underinvestment, market intervention
Email addresses: [email protected], [email protected]. We are grateful to Ulf Axelson, Hans Gersbach,
Arvind Krishnamurthy, Guido Lorenzoni, Semyon Malamud, John Moore, Tyler Muir, Martin Oehmke, Alp Simsek,
Balazs Szentes, Jaume Ventura, Rob Vishny, and numerous seminar participants. We thank Miklós Farkas for
excellent research assistance. Zhiguo He acknowledges the …nancial support from the Center for Research in Security
Prices at the University of Chicago Booth School of Business. Péter Kondor acknowledges the …nancial support of
the Paul Woolley Centre at the LSE and of the European Research Council (Starting Grant #336585).
1
Introduction
The history of modern economies is rich with boom-and-bust patterns. Boom periods during
which vast resources are invested in new projects are followed by downturns during which longrun projects are liquidated early, liquid resources are hoarded in safe short-term assets, and there
is little investment in new projects. While some of these patterns a¤ect only certain industries,1
others a¤ect the aggregate economy— e.g., the emerging market boom and bust at the end of 1990s,
or the recent investment boom around the mid-2000s and the crisis afterwards. These investment
cycles are in the forefront of the academic and policy debate.
In this paper, we show that …rms’ individually optimal liquidity management results in socially ine¢ cient boom-and-bust patterns. Financially constrained …rms choose what level of liquid
resources required to absorb cash-‡ow shocks and to take advantage of time-varying investment opportunities. Firms hold liquid resources both to avoid ine¢ cient liquidation of productive capital in
case of adverse cash-‡ow shocks and to be prepared for potentially cash-intensive future investment
opportunities. Our focus is on the implications for the aggregate economy when cash-‡ow shocks
are correlated across …rms.
Our …rst observation is that …rms’liquidity management decisions generate simultaneous waves
in …rms’ aggregate holdings of liquid assets and investment and waves of the opposite phase in
market value of liquidity, even if technology remains constant. We argue that the emerging picture
partially rationalizes evidence on liquidity holdings of non-…nancial …rms and the time variation in
the market value of liquidity.
The main result of this paper is that we show that such investment waves are not constrained
e¢ cient when future investment opportunities are non-contractible. The social and private value of
liquidity di¤ers in general. In particular, the incentive to turn liquid resources into illiquid capital,
which a¤ects individual …rms but not the planner, is stronger during booms (i.e. after a series of
favorable cash-‡ow shocks so that the capital price is relatively high) than during recessions. We
show that the externality is often two-sided depending on the aggregate state: there is overinvestment in capital during booms and underinvestment in capital during recessions. As a result, …rm
investment is too volatile.
The presence of a two-sided externality radically changes the outcome of policy interventions.
In general, policies targeted on raising prices in recessions help mitigate underinvestment, but make
overinvestment in booms worse. As an example, consider a transfer scheme that does not allow the
price of capital to fall below a certain level during recessions. We show that setting the appropriate
price level for such a policy is critical. If the set price for the recession is not su¢ ciently low, it may
decrease welfare during both booms and recessions, as agents foresee the induced overinvestment
in booms. We show how a speci…c price-‡oor policy can change incentives through all states of the
economy in order to increase welfare during both booms and recessions.
1
For example, Hoberg and Phillips (2010) document a large number of examples of industry-speci…c boom-andbust patterns beyond the well-known examples such as the boom and bust of the semi-conductor industry in the
1990s. (See also Rhodes-Kropf, Robinson and Viswanathan (2005) for related …ndings.)
1
For our analysis, we integrate a novel, analytically tractable, stochastic dynamic model of
liquidity management into a macroeconomic context. Our model focuses on non-…nancial …rms.
We call their long-term risky asset capital, and their liquid asset holdings cash. Capital stands
for certain …xed investment in long-term risky technology, which produces stochastic ‡ows in cash.
Cash can be stored safely, exchanged for consumption goods, or used to build new capital at
a constant proportional cost. Capital can also be liquidated for a relatively smaller constant
proportional bene…t in terms of cash. Thus, aggregate cash holdings represent non-…nancial …rms’
liquid …nancial claims on the rest of the economy. The risky cash ‡ows generated by capital (which
can also be interpreted as short-lived TFP shocks) represent aggregate shocks in our economy, and
negative cash-‡ows imply that capital requires costly maintenance in terms of cash.
The economy is initially in the aggregate stage where identical …rms facing the aggregate cash‡ow shocks trade, build or liquidate capital, or consume. With Poisson intensity, …rms move to
the idiosyncratic stage. In that stage, some …rms …nd a productive project, which uses the existing
capital (capital …rms), while others get a new idea for a project, which requires cash to be exploited
(cash …rms). Then, cash …rms sell their capital to capital …rms in a Walrasian market. After trading,
cash …rms invest all their cash into the new opportunity, whereas capital …rms operate their capital
holdings more productively. Finally, …rms consume all their obtained wealth.
A crucial equilibrium implication of our setup is that the aggregate stage features simultaneous
waves in investment, cash-holding of …rms, and the price of capital in terms of cash, even with
constant technology. Firms store the cash as a bu¤er in order to avoid ine¢ cient liquidation of
capital. As cash-‡ow shocks are perfectly correlated, a series of positive cash-‡ow shocks raise the
aggregate level of cash holding. The larger bu¤er decreases the chance of a series of adverse shocks
forcing …rms to liquidate productive capital, and as a result raises the equilibrium price of capital.
When the price of capital reaches the …xed cost of investment, …rms decide to build new capital.
Analogously, as a result of a series of negative cash-‡ow shocks the price of capital might drop
to the level of the liquidation bene…t, leading …rms to liquidate capital. This process keeps the
aggregate cash-to-capital ratio within the implied liquidation and investment thresholds. We think
of the state when new capital is built as a boom period and the state when capital is liquidated as
a recession.
We show that the equilibrium liquidation and investment thresholds do not coincide with a
planner’s choice if the investment opportunities in the idiosyncratic stage are not contractible. In
the planner’s solution, …rms liquidate their productive capital only when the cash-to-capital ratio
hits zero, and invest during booms when the cash-to-capital ratio hits a positive threshold, which
is the socially optimal cash bu¤er in this economy. However, in the decentralized equilibrium, the
investment and disinvestment thresholds are distorted. In particular, …rms always liquidate capital
at a strictly positive cash-to-capital ratio, implying that …rms always underinvest in downturns.
Interestingly, under some conditions …rms invest in capital when the cash bu¤er is lower than
the one the planner would choose. That is, they underinvest in capital (liquidate too much) in
downturns and overinvest during booms. As a mirror image, they hoard too much cash during a
2
downturn, and hold too little cash during a boom.
Here is the economic intuition: As we noted, …rms’incentive to build liquidity bu¤ers against
cash-‡ow shocks generates procyclicality in aggregate liquidity holdings and countercylicality in the
value of liquidity, implying that the value of capital relative to cash, i.e. the capital price, has to
be procyclical. Once investment opportunities arrive, cash …rms can sell the capital they have, and
capital …rms buy the capital at the prevailing market price in terms of cash. Therefore, in booms,
when the price of capital is higher, …rms value their capital more than cash. That is, preparing
for investment opportunities aggravates procyclicality in capital prices. However, this additional
e¤ect that in‡uences private incentives is absent from social incentives, because one …rm’s gain
from trading capital to cash is the other …rm’s loss. Therefore, there is a state-dependent wedge
between the private and social valuation of capital (relative to cash), creating the possibility of
overinvestment in booms and underinvestment in recessions.
This argument holds because we assume that certain markets are missing. For example, …rms
writing contracts ex ante on investment opportunities would insure each other against the gains
and losses from ex post trading. Similarly, …rms able to pledge the output of their investment
opportunities, would exchange capital to cash at terms determined by the (…xed) output of these
opportunities. These possibilities eliminate the wedge between the market price and the social
value of capital, restoring the constrained e¢ ciency for the decentralized economy.
As an extension of our model, we allow …rms to pledge capital to obtain external credit by
collateralized borrowing. This makes capital more valuable from both the private and social perspectives. We show that collateralized borrowing tends to push up the private bene…t of capital
more than it does on the capital’s social value. Therefore, collateralized borrowing could be excessive, in the sense that a su¢ ciently large borrowing capacity of capital brings a no-borrowing
economy from “underinvestment always” to two-sided ine¢ ciency, with overinvestment during the
boom.
As an illustration of the potential of our mechanism to provide new explanations for existing
problems in various contexts, we connect our results to the observed phenomenon of relative boomand-bust patterns across industries, and to stylized facts that in less …nancially developed countries
investment in productive technologies is more volatile and exhibits stronger procyclicality.
As a methodological contribution, we develop a novel dynamic model to analyze the e¤ect of
aggregate liquidity ‡uctuations on asset prices and real activity, with analytical tractability of the
full joint distribution of states and equilibrium objects.
Literature. In our model, …rms’individually optimal liquidity management decisions generate
aggregate waves in investment, market value of liquidity, and aggregate liquidity holdings. As a
main contribution, we show that if future investment shocks are non-contractible, …rms often have
too much incentives to invest during booms and too little incentives to invest during recessions.
Ours is not the …rst paper to emphasize that …rm-level constraints can generate ine¢ cient investment waves. The literature with perhaps the largest in‡uence on current policy discussions
emphasizes the …re-sale feedback loops induced by a price-sensitive collateral constraint (e.g., Kiy3
otaki and Moore, 1997; Gromb and Vayanos, 2002; Krishnamurthy, 2003; Jeanne and Korinek, 2010;
Bianchi, 2010; Bianchi and Mendoza, 2011; Stein, 2011; He and Krishnamurthy, 2012; Jermann and
Quadrini, 2012; Brunnermeier and Sannikov, 2014). In these models, …rms fail to internalize that
the more they borrow and invest during booms, the more they have to deleverage and disinvest
during recessions, which depresses …re-sale prices and tightens the constraint faced by other …rms as
well. Compared to a social planner facing the same constraints, in these models …rms’incentives to
borrow and/or invest are always too strong. Our research di¤ers from this literature in two crucial
dimensions. First, our mechanism is unrelated to any form of collateral-based or net-worth-based
ampli…cation mechanism. Second, and more important, the externality in our model changes sign
with the state of the economy. As a result, policy measures limiting overexpansion in booms, which
are unambiguously bene…cial in an economy with collateral constraints, cause ine¢ cient hoarding
of liquidity in our economy and potentially decrease welfare everywhere.2
Like the literature on …re-sale feedback loops, our work also belongs to the literature analyzing
the welfare e¤ects of pecuniary externalities. This literature is based on the seminal papers of
Stiglitz (1982), Greenwald and Stiglitz (1986), and Geanakoplos and Polemarchakis (1985), which,
like the recent work of Farhi and Werning (2013), establish general conditions implying welfarechanging pecuniary externalities. Our application of this general principle is closest to the vein of
research in which market incompleteness hinders the equalization of …rms’marginal utility of wealth
across states or time (e.g., Shleifer and Vishny (1992), Allen and Gale (1994, 2004, 2005), Caballero
and Krishnamurthy (2001, 2003), Lorenzoni (2008), Farhi, Golosov and Tsyvinski (2009) and Gale
and Yorulmazer (2011)). Compared to a planner, this mechanism can imply that incentives to invest
are either too strong or too weak, depending on the exact speci…cation.3 Our main innovation is that
we highlight the e¤ect of interacting these types of pecuniary externalities with varying incentives
to hold liquid assets over the cycle. This interaction leads to our main result that the sign of the
distortion in investment incentives switches with the state of the economy.
A few recent papers cast in two-period settings investigate two-sided ine¢ ciency and derive
implications related to our work. Gersbach and Rochet (2012) study the moral hazard problem of
incentivizing banks in a macroeconomic context, and show that banks extend too much credit in
booms and too little in recessions. Their mechanism relies on the di¤erence between the private
and social solution of bank’s moral hazard problem. Additionally, in their two-period setting which
models booms and recessions separately as two di¤erent states in period 1, the period 0 intervention
can resolve the two-sided e¢ ciency at once. In contrast, in our dynamic model booms and recessions
2
This paper contributes to the discussion on the optimal mix of ex ante regulation and ex post intervention (e.g.,
Diamond and Rajan, 2011; Farhi and Tirole, 2012; Jeanne and Korinek, 2013), to the extent that we emphasize that
a policy of intervention during a recession will also a¤ect incentives during a boom. We characterize economies when,
because of the two-sided externality, this fact has crucial consequences on the welfare e¤ects of these policies.
3
See Davila (2014) for a comparative analysis of di¤erent mechanisms connected to pecuniary externalities and
the argument that collateral constraints always imply overinvestment ex ante. For uninsurable idiosyncratic liquidity
shocks, see Holmstrom and Tirole (2011, chap.7.) of simpli…ed versions and excellent discussion of Shleifer and Vishny
(1992) and Caballero and Krishnamurthy (2003). Finally, a recent paper by Hart and Zingales (2011) studies the
excessive supply of private money based on the idea of special pledgeability of certain assets. This friction always
results in overinvestment in such assets, in contrast to our model.
4
occur in cycles, and the potentially inferior one-sided interventions emphasize the interconnected
incentives between booms and recessions for forward-looking economic agents. Eisenbach (2013)
studies banks …nanced with short-term debt in a general equilibrium setting, and show that in
good (bad) times banks face too little (much) market discipline imposed by rolling over short-term
debt. In contrast to our paper, in which idiosyncratic investment opportunities drive ine¢ ciency,
that paper emphasizes aggregate risk, and the fact that short-term debt lacks aggregate-state
contingency.
In our model, …rms hold liquid assets to avoid adverse e¤ects of cash-‡ow shocks and to prepare
for future investment opportunities. This is consistent with a large body of previous work on
liquidity management (e.g., Almeida, Campello and Weisbach (2004), Bates, Kahle and Stulz
(2009), Denis and Sibilkov (2010), Ivashina and Scharfstein (2010), Lins, Servaes and Tufano (2010),
Eisfeldt and Muir (2013), Acharya, Almeida and Campello (2013)). This argument goes back to
Keynes, who calls this the precautionary motive.4 However, instead of aiming for a detailed picture
of …rms’individual saving and investment decisions, we focus on the consequences of such decisions
to the aggregate economy.
The structure of our paper is as follows. In Section 2 we present the setup and the equilibrium
of our model. In Section 3 we expose the ine¢ ciencies of the market solution. Section 4 presents our
…ndings on economic policy and other applications. We discuss the robustness of our mechanism in
Section 5. We conclude in Section 6. All proofs are in Appendix, Online Appendix, or Additional
Material available on the author’s website.5
2
A Dynamic Model of Saving and Investment
2.1
Assets
We model an economy where …rms facing cash-‡ow shocks and time-varying investment opportunities make saving and investment decisions. There is a single capital good representing risky and
productive projects. The other asset in this economy is cash which serves both as a consumption
good and as an input for building capital. We assume that there is a safe storage technology and
that capital does not depreciate; thus both capital and cash are perfectly storable.
For each …rm, there is a …nal date arriving at a stopping time
where
with Poisson intensity ,
is a positive constant. At this …nal date, …rms receive potentially di¤erent investment
opportunities (to be speci…ed shortly), and any unused capital depreciates fully. For now, we think
of the arrival of the …nal date as an aggregate shock (we o¤er an alternative interpretation in Section
2.4). Before the …nal date, each unit of capital generates random cash ‡ows. This shock is common
across capital units and driven by dZt , where
is a positive constant and Z
fZt ; Ft ; 0
t < 1g
is a standard Brownian-motion on a complete probability space ( ; F; P). One can interpret the
4
Others proposed the tax motive, the transaction motive, and the agency motive as alternative explanations (see
Bates, Kahle and Stulz (2009) for detailed arguments and references).
5
See http://faculty.chicagobooth.edu/zhiguo.he/research/hekondor_additionalmaterial.pdf.
5
aggregate cash-‡ow shocks dZt as short-lived TFP shocks. When dZt > 0 the capital generates
cash. When dZt < 0; the …rm needs to spend j dZt j amount of cash on this capital as maintenance
cost; otherwise the capital turns unproductive.
Denote by Kt the aggregate quantity of capital. Given the aggregate cash shock dZt of each
unit of capital, when …rms do not invest or disinvest (to be introduced shortly), the aggregate level
of cash accumulated in storage, Ct ; would follow the evolution of
dCt = Kt dZt :
2.2
(1)
Firms and frictions
The market is populated by a unit mass of risk-neutral …rms who operate the capital. At each time
instant, …rms may decide to build new capital, trade capital for cash at the equilibrium price pt , or
liquidate the capital. Building new capital costs h units of cash, while liquidating a unit of capital
provides l units of cash, where h > l > 0. Firms can also consume their cash at any moment of
a constant marginal utility of 1. Because of linear technologies, in general it is optimal to have
threshold strategies of (dis)investment. Thus, we can simply focus on thresholds in comparing
di¤erent (dis)investment strategies.
The major friction in this economy is that …rms can neither write contracts on the di¤erent
investment opportunities they face, nor they can pledge the future return on these opportunities.
Although …rms are initially identical, they receive di¤erent investment opportunities. Speci…cally,
in the random …nal date, each …rm with probability half …nds a project which uses the existing
capital productively generating RK > 0 unit of …nal consumption per each unit of used capital.
The other group of …rms …nd a new idea requiring liquid resources. Hence, this latter group have a
superior use for liquid resources, and we assume that they receive RC > 1 unit of …nal consumption
per unit of cash invested. These shocks are independent across …rms, and we refer to the earlier
group as capital …rms and the latter group as cash …rms. RK and RC are positive constants. Our
extreme assumption that neither group’s project returns are pledgeable is a short-cut for agency
and/or informational frictions.6 We partially relax this assumption in Section 5.2. Throughout we
assume that
RK
> h;
RC
(2)
which ensures that building capital is socially e¢ cient when the economy has su¢ cient cash.7
Firms learn which group they belong to only at the beginning of the …nal date. In the …nal date
the conversion technology between capital and cash is no longer available, but …rms have a last
trading opportunity to trade capital for cash before …nal production. We refer to the potentially
in…nitely long interval before the …nal date
as the aggregate stage of the economy, as at this stage
6
Appendix C of He and Kondor (2012), in the context of a simple two-period example, discusses the potential
agency problems in detail.
7
K
Allowing for h > R
> l would leave the derivation and the characterization of the market equilbirium untouched.
RC
Although the comparision to the planner’s case remains simlar, the derivation is more cumbersome. Hence, for easier
readability, we discuss this case in Remark 1 in Section 3.2.
6
Figure 1: Time line.
all shocks a¤ect each agent the same way. By similar logic, we refer to the …nal date
(in which
…nal trading occurs) as the idiosyncratic stage. We denote the price in the idiosyncratic stage by
pb (recall that we denote by pt the prices in the aggregate stage). Figure 1 summarizes the time
line of events in our model. We expand on the interpretation of the two stages in Section 2.4.
2.3
Individual …rm’s problem
Consider …rm i, which holds Kti units of capital and Cti amount of cash, with a wealth (in terms
of cash) of wti
pt Kti + Cti : Since the idiosyncratic stage arrives according to an exponential
distribution with density e
fd
i
where
d
i
t
max
0;K i 0;C i 0;dK i g
i
t
E
Z
, …rm i is solving the following problem:
1
e
0
Z
d
0
i
t
+
1
2
Ki +
Ci
pb
RK +
is …rm i’s cumulative consumption before the …nal date
0; later we see that it is zero in equilibrium), and
dKti
1
K i pb + C i RC
2
d
(3)
(so it is non-decreasing with
is the amount of capital that it
dismantles or builds. The term in the squared bracket is the consumption at the idiosyncratic
stage. For instance, if the …rm turns out to be cash-type, it will sell its capital holding K i at
the price of pb to receive K i pb , and then invest its cash together with C i in exploiting new cashintensive projects with return RC .
The problem in (3) is subject to the dynamics of individual wealth,
dwti =
where
d
i
t
dKti + Kti (dpt + dZt ) ;
is the cost of changing the amount of capital, so that
wti
= h1fdK i
t
(4)
0g
+ l1fdK i <0g . Also,
t
wealth cannot be negative at any point, i.e.
0 of all t:
R i
Recall Kt = i Kt di is the aggregate capital. Combining the investment/disinvestment policy
7
dKt , (1) implies that the dynamics of aggregate cash level in the economy is8
dCt = Kt dZt
dKt :
(5)
The scale-invariance implied by the linear technology suggests that it is su¢ cient to keep track of
the dynamics of the cash-to-capital ratio:
Ct
;
Kt
ct
which evolves according to
dct =
2.4
dCt
Kt
Ct dKt
= dZt
Kt Kt
( + ct )
dKt
:
Kt
(6)
Interpretation of the aggregate and idiosyncratic stages
We stress that thinking of the arrival of the idiosyncratic stage as an aggregate shock and the
resulting separation of the two stages is a didactic tool. It helps show how the incentives related
to the idiosyncratic investment opportunities a¤ect the incentives for saving and investing in the
aggregate stage. In the real world, some …rms might be in the idiosyncratic stage while others
are still in the aggregate stage. Therefore, the …nal date does not correspond to an observable
time point in the economy. Instead, we will think of recessions and booms and economic policies
a¤ecting saving and investment in these states within the aggregate stage of the economy. With
this structure we can analyze the dynamic ‡uctuation of our economy without sacri…cing analytical
tractability.
Indeed, there is a formally equivalent economy where the arrival of the …nal date is idiosyncratic
to individual …rms. Under this interpretation, in each time interval dt a dt fraction of …rms
randomly receive heterogenous investment opportunities as above (i.e.,
…rms while the other
2 dt
2 dt
fraction are capital
fraction cash …rms), enter the idiosyncratic stage and trade cash for
capital among themselves on a separate market, while the remaining …rms continue to operate in
the aggregate stage. Thus, under this interpretation the economy never terminates.
In this alternative economy, the individual …rm’s problem (3) and the evolution of aggregate
state (6) remain the same. Because at each instant there are equal fractions of cash and capital
‡owing out from the economy, the aggregate cash-to-capital ratio in the remaining economy is not
a¤ected; but the size of the remaining economy shrinks. The trading price also remains pb in the
separate market, while all incumbent …rms face a trading price of pt .
To further emphasize that this separation is a technical innovation, in Section 5.1 we present and
analyze a version of our model where a fraction of …rms learn about new investment opportunities
in each time instant and trade cash and capital in a single market together with the …rms who
remain in the aggregate stage. That is, the aggregate stage and the idiosyncratic stage are not
8
To simplify notation we ignore the possibility that at any given point in time some …rms create capital while
some …rms liquidate capital. It is easy to see that this never happens in equilibrium.
8
separated. While that version is not analytically tractable, we will illustrate by numerical analysis
that our main result goes through.
2.5
Interpretation of cash and capital
Cash holding in the aggregate stage, Cti ; represents the …nancial slack of a …rm— cash holdings;
other short-term, liquid investments; or credit lines. It can be used either to cover any operating
losses, or invest in any new opportunities (even outside the industry). As we illustrate in Figure
3, it is possible to map Cti to data by thinking of it as the liquid …nancial asset holdings of non…nancial …rms of the economy. Note that in reality these assets represent claims on the government,
households, or foreigners: entities which we do not explicitly model.
Kti represents …rms’total gross property, plants, equipment, inventories, and intangible assets,
which is much more speci…c to each industry and thus much less liquid. The process
Kti dZt
might represent cash ‡ows from both operating and …nancing activities. In our abstract model
without external …nancing, …rms …nance their investment from retained earnings only. However, in
Section 5.2 we show that allowing for collateralized borrowing could make our main results more
pronounced.
Importantly, RC in the idiosyncratic stage should not be interpreted as the return from liquid
investments. Instead, it is a reduced-form representation of the expected return from the cashintensive development of a new idea. We follow a reduced-form treatment. In reality, the cash
might have to be used to hire labor, or purchase speci…c capital for the new idea. RK can be
interpreted similarly, but for an idea that uses the same type of capital as the existing technology.
Cash …rms are the ones with comparative advantage in exploiting the former, whereas capital …rms
have comparative advantage in exploiting the latter.
2.6
De…nition of equilibrium
De…nition 1 In the market equilibrium,
1. each …rm chooses d it ; Kti ; Cti ; and dKti to solve (3), and
2. markets clear in every instant, during both the aggregate and the idiosyncratic stages.
As we will see, in our framework, the equilibrium only pins down the aggregate variables: prices,
net trade, and net investment and disinvestment. Typically, any combination of individual actions
consistent with the aggregate variables is an equilibrium. It is convenient to pick the particular
market equilibrium where all …rms follow the same action, which we refer to as the symmetric
equilibrium.
Henceforth, we omit the time subscript t or
whenever it does not cause any confusion.
9
2.7
Market equilibrium
We solve for the market equilibrium in this section. As we show, in this economy consumption (of
cash) before the idiosyncratic stage is strictly suboptimal, thus d
2.7.1
i
t
=d
t
= 0 always.
Equilibrium price in the idiosyncratic stage
Consider the idiosyncratic stage. The law of large numbers implies that there exists a half measure
of capital (cash) …rms. All capital …rms use their cash holdings to buy capital holdings from cash
…rms, and the market clearing condition implies that
We still need to ensure that RK
1
1
C = K pb ) pb = c:
2
2
pb = c: This is because capital …rms have the option of consuming
their cash holdings instead of purchasing capital, which puts an upper bound on pb. Later we show
that the full support of c is endogenous, because …rms build (dismantle) capital whenever the
aggregate cash is su¢ ciently high (low). For simplicity, we restrict the parameter space to ensure
that the condition c
2.7.2
RK holds always in equilibrium.
Equilibrium values, prices, and investment polices in the aggregate stage
Now we determine equilibrium objects in the aggregate stage. The next lemma states two useful
features of our formalization: First, the only relevant aggregate state variable is the cash-to-capital
ratio. Second, the value function of any individual …rm is linear in its capital and cash holdings.
Lemma 1 Let J K i ; C i ; K; C be the value function of …rm i which holds capital K i and cash C i
in an economy with aggregate capital K and aggregate cash C. Then, for aggregate cash-to-capital
ratio c = C=K; there are functions v (c) and q (c) that,
J C; K; K i ; C i = K i v (c) + C i q (c) :
That is, regardless of the …rm’s composition of asset holdings, the value of every unit of capital
is v (c), and the value of every unit of cash is q (c). Both functions depend only on the aggregate
cash-to-capital ratio. Because of linearity, the equilibrium price has to adjust in a way such that
…rms are indi¤erent to whether they hold capital or cash. That is, the equilibrium price of capital
p (c) in the aggregate stage must satisfy that
p (c) =
v (c)
:
q (c)
Firms build capital whenever the capital price p reaches the cash cost h; and they dismantle
capital whenever the price falls to the liquidation bene…t l: De…ne ch and cl as the endogenous
thresholds of the aggregate cash-to-capital ratio where …rms start to build and dismantle capital,
10
respectively. These thresholds satisfy
v (cl )
v (ch )
= h; and
= l:
q ch
q cl
(7)
Moreover, the linear technology implies that ch and cl are re‡ective boundaries of the process c.
Therefore, based on (6), the aggregate cash-to-capital ratio c must ‡uctuate in the interval [cl ; ch ],
with a dynamics of
dc = dZt
where dUt
dUt + dBt ;
(8)
t
(h + ch ) dK
Kt re‡ects c at ch from above, while dBt
t
(l + cl ) dK
Kt re‡ects c at cl
from below. The standard properties of re‡ective boundaries imply the following smooth-pasting
conditions for our value functions (Dixit (1993)):
v 0 (ch ) = q 0 (ch ) = q 0 (cl ) = v 0 (cl ) = 0:
2.7.3
(9)
Characterizing the market equilibrium
Now we turn to characterizing the value functions v (c) and q (c) in the range c 2 [cl ; ch ] : Here we
give a sketch; full details are available in the Online Appendix. Because of Lemma 1, …rms are
indi¤erent to the composition of their asset holdings, and we can consider the value function of a
…rm that holds only capital or only cash. The value function of a …rm holding only cash gives an
Ordinary Di¤erential Equation (ODE) of q (c):
2
0=
q 00 (c)
2
| {z }
volatility of dct
+
|2
(RC
q (c)) +
2
{z
}
|
becoming a cash …rm
RK
q (c) ;
c
{z
}
(10)
becoming a capital …rm
and the value function of a …rm holding only capital, given q (c) ; yields the ODE for v (c):
2
0=
v 00 (c)
2
| {z }
volatility of dct
+
q 0 (c) 2
| {z }
+
expected value of dividends
|2
(RC c v (c)) +
{z
}
becoming a cash …rm
|2
(RK
{z
v (c)) :
}
(11)
becoming a capital …rm
These ODEs are Hamilton-Jacobi-Bellman (HJB) equations for cash and capital given the
dynamics of the state c. We …rst explain the terms unrelated to
correction term
2
2
q 00 (c)
in each ODE. For (10), the Ito
captures the impact of the evolution of the state variable c; a similar term
shows up in (11). In addition, we have q 0 (c)
2
in (11) because the capital generates random cash
‡ows dZt which are perfectly correlated with the aggregate state ct+dt = ct + dZt (see (8)).9
Multiplied by the intensity , the terms describe the change in expected utility once the idiosyncratic stage arrives. The …rst of these terms in (10) captures that, once a …rm holding a unit
of cash learns to be a cash …rm, its value jumps to RC from q (c) : The second term says that it
9
Heuristically, given q ( ) as the marginal value of cash, the expected value of the cash ‡ows dZt standing at time
t is Et [q (c + dZt ) dZt ] = Et q 0 (c) 2 (dZt )2 = q 0 (c) 2 dt:
11
uses the unit of cash to buy 1=b
p = 1=c unit of capital, so its value jumps to RK =c from q (c) : The
interpretation in (11) is analogous.
p
De…ne the constant
2 = . The ODE system in (10)-(11) has the closed-form solution:
q (c) =
RC
+e
2
c
A1 + ec A2 + RK
ec Ei (
2
c) + e
2
c
Ei (c )
;
(12)
and
v (c) = RK +
where Ei (x)
(e c Ei ( c) e c Ei ( c))
RC c
+ ec (A3 cA2 ) e c (A4 + cA1 ) + cRK
; (13)
2
2
2
Rx
1 t
1 t e dt is the exponential integral function, and the constants A1 -A4 are
determined from boundary conditions in (9).
Finally, we determine the endogenous investment/liquidation thresholds cl and ch using (7). The
functions v (c) ; q (c) and the thresholds constitute an equilibrium if the resulting price p (c) =
v(c)
q(c)
falls in the range of [l; h] when c 2 [cl ; ch ]. The following proposition gives su¢ cient conditions for
such a market equilibrium to exist and describes the basic properties of this equilibrium.10
Proposition 1 If the di¤ erence between the bene…t of liquidation, l, and the cost of building capital,
h, is su¢ ciently small, then the market equilibrium exists with the following properties:
1. …rms do not consume before the …nal date;
2. each …rm in each state c 2 [cl ; ch ] is indi¤ erent to the composition of its asset holdings and
0 < cl < ch < RK ;
3. …rms do not build or dismantle capital when c 2 (cl ; ch ) and, in aggregate, …rms spend every
positive cash shock to build capital if and only if c = ch , and they cover negative cash shocks
by liquidating a su¢ cient fraction of capital if and only if c = cl ;
4. the value of holding a unit of cash and the value of holding a unit of capital are described by
v (c) and q (c), and the price in the aggregate stage is p (c) = v (c) =q (c);
5. in the idiosyncratic stage, a capital …rm sells all its capital to cash …rms for the price pb (c) = c;
6. q (c) is monotonically decreasing, v (c) is monotonically increasing, and p (c) is monotonically
increasing. Furthermore, q (c) has exactly one in‡ection point: there is a cq 2 (ch ; cl ) such
that q 00 (c) < 0 for c 2 (cl ; cq ) and q 00 (c) > 0 for c 2 (cq ; ch ).
2.7.4
Investment waves
The thick, solid curves on panels A-E of Figure 2 illustrate the properties of the market equilibrium. In panels A-C, the functions p (c) ; v (c) ; q (c) describe the price of capital, the value of
10
When h l is not su¢ ciently small, a variant of this equilibrium often prevails. Because this variant has very
similar features, we relegate the discussion of it to Additional Material, available on the author’s website.
12
cash, and the value of capital, respectively. Panels D-E depict the cash-to-capital ratio and the
investment/disinvestment activity along one particular sample path.
The cash-to-capital ratio, c; represents the relative scarcity of liquid assets in the economy
compared to illiquid capital. Thus, we refer to this ratio as “aggregate liquidity.” We also think of
intervals with a large increase (drop) of capital as a boom (downturn). In our model, investment
takes a simple threshold strategy, in such a way that investment (disinvestment) occurs only at ch
(cl ). However, we believe the resulting clustered investment and disinvestment activities depicted
in panel E captures the essence of boom-and-bust patterns observed in reality.
The economy ‡uctuates across states because the aggregate cash-‡ow shocks drive the level of
aggregate liquidity. This is illustrated in panel D. This particular sample path starts with a series
of positive shocks, which increase the capital value v and decrease the cash value q. Thus, the price
of capital increases along this path (not shown),11 because in these states the probability that the
economy will slip into a downturn (and capital must be dismantled) is low. When the price hike
reaches the cost of building capital, h; investment is triggered (as shown in Panel E). This keeps
the cash-to-capital ratio below ch : For symmetric reasons, as a series of negative shocks decrease
aggregate liquidity, rising cash values and falling capital values lead to lower capital prices. When
the price of capital drops to l, disinvestment in capital is triggered. This keeps the cash-to-capital
ratio above cl .
Figure 3 shows our …rst step in mapping our model to data. Based on FED Flow of Funds data,
we construct a series of aggregate liquid …nancial assets for non-…nancial US-…rms, normalized by
the nominal GDP, and showing NBER recessions as shaded areas. Based on the FRED database,
we also plot the CD/T-bill spread as a proxy for the market value of liquidity; this spread is often
used to measure the liquidity premium as CD is relatively less liquid compared to T-Bills. We also
show the cyclical component of both series. These two series correspond to aggregate liquidity, ct ;
and the value of a unit of liquidity, q (ct ) in our model. In the data, the cyclical components of the
two series are negatively correlated, with a coe¢ cient of
0:3.
Note that in recessions, liquid …nancial assets tend to be low but the value of liquidity tends
to be high. Indeed, the correlation between the cyclical component of liquid …nancial assets and
the recession dummy is
0:5. These observations support our interpretation that recessions are
associated with relatively low aggregate holdings of liquid assets and high valuations for liquidity.
As we will explain in the rest of the paper, the general pattern of investment waves, procyclical
liquidity holdings, and countercyclical valuation for liquidity are a robust pattern in our economy.
These features are present regardless of whether the economy is constrained e¢ cient. It turns out
that the e¢ ciency properties of our economy are determined by whether the investment thresholds
cl and ch are at their welfare-maximizing level. We examine this issue in the next section.
11
As a monotonically increasing function of c; the path of p (c) looks qualitatively similar to the path of c, except
that it ‡uctuates between h and l instead of ch and cl .
13
Figure 2: Panels A-C depict the price of capital, the value of cash, and the value of capital. The solid vertical
line on the right of each graph is at the investment threshold in the planner’s solution, cP
h = 4:03; while the two
dashed vertical lines are the disinvestment and investment thresholds in our baseline case, cl = 1:13; ch = 3:14: The
horizontal lines on Panel A are at h and l: Panels D-F depict a simulated sample path. Horizontal lines on panel D
from top to bottom are cP ; ch and cl : Each panel shows objects of both the baseline model with competitive market
(thick solid curves) and the planner’s solution (thin, dashed curves). Parameter values are RK = 4:2; RC = 2,
2
= 0:6; = 0:1; l = 1:8 and h = 2:
14
Figure 3: Quarterly aggregate liquid …nancial assets for non-…nancial US-…rms normalized by nominal GDP (calculated as the sum of items 1-14 in Flow of Funds Tables L.102, with NBER recessions as shaded areas); and CD/T-bill
spread as a proxy for the market value of liquidity (CD6M/TB6M series in FRED database). Panel A plots the raw
series, and Panel B plots the cyclical component applying the Baxter-King …lter.
3
Welfare
To study pecuniary externalities, we …rst solve for constrained e¢ cient allocation in this economy
as a benchmark. We then show that our model features a two-sided ine¢ ciency on investment
waves: Firms underinvest in capital during downturns and often overinvest during booms.
3.1
Constrained e¢ cient benchmark
We study the constrained e¢ cient allocation with the technological constraint that the aggregate
cash has to be kept non-negative by liquidating capital if necessary. Without this technological
constraint, condition (2) implies that the planner should convert any amount of cash to capital.
We consider a social planner who can dictate investment policies but cannot know the realization
of the idiosyncratic shock. Compared to the market equilibrium, the only di¤erence is that in the
market equilibrium investment and disinvestment are driven by the market price of capital. In
contrast, the social planner ignores market prices and directly decides when to build or dismantle
capital. The resulting outcome corresponds to the solution of the planner’s problem when he
controls both investment in the aggregate stage and allocation in the idiosyncratic stage, given
self-reporting (see He and Kondor (2012) for a detailed argument).
3.1.1
Social planner’s problem
Denote by JP (K; C) the planner’s value function which decides when to build and dismantle capital.
By the end of the idiosyncratic stage, at least as long as ct
15
RK ; due to linearity all cash ends up
in the hands of cash …rms and all capital ends up in the hands of capital …rms.12 Therefore, the
total value in the idiosyncratic stage is13
KRK + CRC :
(14)
Thus, given the aggregate state pair (K; C), since the …nal date
arrives with exponential distrib-
ution with intensity , the social planner is solving
JP (K; C) = max E
dK
Z
1
e
(K RK + C RC ) d
K0 = K; C0 = C
KjP
0
C
K
= KjP (c)
(15)
subject to the constraint Ct
0 and (5). In the second equality in (15), we have invoked the
scale-invariance to de…ne jP (c) as the planner’s value per unit of capital.
Because of the linear technology, regulation with re‡ective barriers on c is optimal (Dixit (1993)).
That is, there exists lower and upper thresholds cPl
0 and cPh > cPl , so that it is optimal to stay
inactive whenever c 2 cPl ; cPh , and dismantle (build) just enough capital to keep c = cPl (c = cPh ).
Consider a given policy fcl ; ch g in which c is regulated by re‡ecting barriers cl < ch . Given
initial state K0 = K and C0 = cK, de…ne the corresponding (scaled) social value as jP (c; cl ; ch ),
so that
K jP (c; cl ; ch )
E
Z
1
e
(K RK + C RC ) d
K0 = K; C0 = cK; cl ; ch :
(16)
0
Using standard results in regulated Brownian motions, jP (c) must satisfy
2
0=
2
jP00 (c) + (RK + RC c
jP (c)) ; for c 2 (cl ; ch ) ;
(17)
and at the re‡ective barriers cl ; ch the smooth pasting conditions must hold:
@ [KjP (cl ; cl ; ch )]
@ [KjP (cl ; cl ; ch )]
@ [KjP (ch ; cl ; ch )]
@ [KjP (ch ; cl ; ch )]
=l
; and
=h
:
@K
@C
@K
@C
(18)
We emphasize that these conditions are not optimality conditions. They hold for any arbitrarily
chosen barriers cl < ch as a consequence of forming expectations on a regulated Brownian-motion
12
This result relies on the linearity of technology and can be formally shown by the mechanism design approach
(see Additional Material). Also, the conditions of Proposition 1 ensure that in the decentralized case p^
ch < RK ;
therefore capital …rms are willing to use all their cash to buy capital, instead of consuming their cash. However, in
the planner’s solution, even for the same parameter values, it might be the case that the support of ct is not a subset
of [0; RK ]. Then, the planner who does not know idiosyncratic …rm types cannot ensure that only cash …rms are
the end users of all cash. While our Propositions 2-6 are stated for the general case, we limit the discussion in the
main text to the simpler case when ct 2 [0; RK ] in the planner’s solution. We show that the Propositions hold in the
remaining cases in Additional Material by explicitly solving the planner’s problem based on the mechanism design
approach when ct > RK has a positive support.
13
Given (K; C), the representative cash …rm gets RC (b
pK + C) = RC (cK + C) = 2CRC ; while the representative
capital …rm gets RK K + RK C= (b
p) = RK K + RcK C = 2KRK : As both types are equally likley the expected total
welfare is KRK + CRC .
16
(see Dixit (1993)). The ODE (17) has a closed-form solution
c
jP (c; cl ; ch ) = RK + RC c + D1 e
+ D2 e c :
(19)
For any …xed fcl ; ch g, we solve for the constants D1 , D2 based on (18).
Denote by cPl ; cPh
the social planner’s optimal barrier pair. With a slight abuse of notation,
we denote the planner’s optimal value, jP c; cPl ; cPh , simply by jP (c):
jP (c)
jP c; cPl ; cPh = max jP (c; cl ; ch ) :
(20)
cl ;ch
Following Dumas (1991), we impose super-contact conditions to determine the optimal barrier pair.
For the upper barrier cPh , this is
@ 2 KjP C=K; cPl ; cPh
@K@C
=h
@ 2 KjP C=K; cPl ; cPh
(@C)2
C=KcP
h
For the lower barrier cPl , at the optimal choice the constraint C
super-contact condition is a complementarity slackness
@ 2 KjP C=K; cPl ; cPh
@K@C
l
(21)
0 might bind. Thus, the
condition14
@ 2 KjP C=K; cPl ; cPh
C=KcP
l
:
C=KcP
h
(@C)2
; with equality if cPl > 0
C=KcP
l
(22)
The next proposition shows that the optimal lower threshold is
cPl
= 0. However, the optimal
upper threshold is characterized by the unique solution to an analytical equation. We explain the
intuition in Section 3.1.3.
Proposition 2 The planner dismantles capital whenever c reaches cPl = 0 and builds capital whenever c reaches a …nite, strictly positive investment threshold cPh : When the unique solution to the
following equation
RK
RK
hRC cP
e h (1 + l )
lRC
(1
l )e
cP
h
2
cPh + h = 0
(23)
lies in [0; RK ], this solution is the socially optimal investment threshold: The optimal social value
jP (c) is concave over 0; cPh .
While the market price in the aggregate stage is unde…ned in an economy where the social
planner sets the investment and disinvestment thresholds, we can de…ne the shadow price of capital,
pP (c) ; as the ratio of the planner’s marginal valuation of capital,
14
@JP (K;C)
,
@K
over that of cash,
Heuristically, we can understand the super-contact condition as follows. Converting capital to cash at a cost of l
P
brings a marginal gain of JK cP
l K; K + lJC cl K; K , and the social planner is considering the marginal impact of
P
P
reducing cl on this marginal gain, i.e., JKC cl K; K
lJCC cP
l K; K . At the optimal policy this marginal impact
P
is zero. If the optimal policy is binding at cl = 0, then this marginal bene…t of reducing cP
l remains strictly positive.
17
@JP (K;C)
,
@C
where
@JP (K; C)
@JP (K; C)
= jP0 (c) ;
= jP (c)
@C
@K
cjP0 (c) ; pP (c) =
jP (c) cjP0 (c)
:
jP0 (c)
(24)
We plot these objects in Figure 2 along with market equilibrium counterparts.
3.1.2
Investment thresholds, welfare, and expected investment volatility
As a preparation for our welfare analysis, we show that (scaled) social welfare, jP (c; cl ; ch ) ; is
monotonic in thresholds in the following sense: It is welfare improving to decrease the lower threshold (increase the upper threshold), whenever it is above (below) the choice of the the social planner.
This is a strong global result: First, it holds for any policy pair as long as cl > 0 and ch < cPh .
Second, the sign of welfare impact by changing investment thresholds is unambiguous everywhere.
Proposition 3 For any ch < cPh and cl > 0, we have
@jP (c; cl ; ch )
@jP (c; cl ; ch )
< 0, and
> 0 for all c 2 [cl ; ch ] :
@cl
@ch
It is also useful to de…ne a measure of the volatility of our investment waves. For this purpose,
we de…ne the expected total adjustment of capital, parameterized by the thresholds cl ; ch :
T (c; cl ; ch )
E
Z
0
jdKt j
:
Kt
(25)
Proposition 4 For any ch and cl , we have
@T (c; cl ; ch )
@T (c; cl ; ch )
> 0; and
< 0:
@cl
@ch
This proposition states that the expected investment volatility increases in the disinvestment
threshold, cl ; and decreases in the investment threshold, ch . Thus, if in the market equilibrium
ch < cPh and cl > 0, then the economy exhibits more volatile investment compared with that in the
constrained e¢ cient benchmark.
3.1.3
Investment thresholds in market equilibrium and in the planner’s solution:
intuition and comparative statics
As the welfare properties of our economy can be traced back to the investment thresholds, it is useful
to understand the economic forces that determine them. As we have established in Propositions 1
and 2, the disinvestment threshold in the market equilibrium, cl > 0; is strictly positive, whereas
the planner disinvests only when it is unavoidable, cPl = 0: In the next proposition, we state further
results and then proceed to the intuition.
Proposition 5 The following results hold.
18
P
Figure 4: Investment and disinvestment thresholds for the planner (cP
l = 0, ch in dashed line) and for the market
2
= 0:6; = 0:1; l = 1:8 and h = 2:
(cl in dotted line, ch in solid line). Parameters are RK = 4:2; RC = 2,
1. The solution of equation (23) determining the planner’s investment threshold, cPh
(a) is converging to 0 as
! 1; and decreasing in
given that
> ^ for a given ^ ;
(b) is decreasing in l and RK , and increasing in h and RC .
2. In contrast, in the market equilibrium determined in Proposition 1, we have
(a) ch > h and cl < l,
(b) ch ! h and cl ! l as
! 1:
The planner starts disinvesting only when he is forced to, i.e., cPl = 0. Intuitively, the planner
does not want to dismantle capital as long as he has not run out of cash yet. A positive lower
threshold would imply that a part of the cash bu¤er is never used for maintenance. Because capital
is more productive than cash, that would be a waste. His choice of the investment threshold cPh > 0
is driven by a simple trade-o¤. While capital is more productive than cash, a cash bu¤er is useful
to avoid the ine¢ cient liquidation of capital in the case of a series of adverse cash-‡ow shocks.
p
Consider the role of the constant = 2 = . This parameter enters (23), which characterizes
the constrained e¢ cient solution, as well as the functions q (c) ; v (c) in (12) and (13), which
characterize the market equilibrium. Figure 4 plots the planner’s investment threshold cPh (dashed)
as a function of .
Intuitively,
measures the relative importance of aggregate cash-‡ow shocks to idiosyncratic
investment opportunities. When
is large, aggregate shocks are less important, either because
their volatility is low, or because the idiosyncratic shock arrives with high intensity. Regardless of
the particular reason, a larger
implies that the planner puts less weight on the possibility that a
19
sequence of negative cash-‡ow shocks force him to dismantle capital at the lower threshold. In fact,
as Proposition 5 states, as
increases without bound, cPh converges to zero as the planner decides
not to store any cash (i.e., he will immediately convert any cash to capital) given that capital is
relatively more productive RK > hRC . Figure 4 illustrates that the smaller the
(say, the larger
the cash-‡ow volatility ), the more weight the planner puts on the possibility of forced liquidation,
and the larger cash bu¤er the planner wants to keep.
Figure 4 also plots the investment thresholds ch (solid) and the disinvestment threshold cl
(dotted) for the market equilibrium. In the market solution, the same trade-o¤ is present, which is
behind the fact that ch is decreasing in
(just as cPh does). However, there is an additional force:
The …rm in the market equilibrium knows that in the idiosyncratic stage the price of capital will be
p^ = c : A ct close to 0 implies that holding on to a bit of cash is a good idea because a small amount
of cash can be exchanged for a large amount of capital in case the economy enters the idiosyncratic
stage. (From the social perspective, the losses and gains from trade in the idiosyncratic stage are
a wash.) Hence, the …rm liquidates capital well before negative cash-‡ow shocks deplete all the
capital stock, implying that cl is bounded away from 0. That is, the reason to liquidate capital
before all cash is depleted is to turn this unit of capital to l units of cash in the aggregate stage,
instead of p^ = c units of cash in the idiosyncratic stage. Clearly, this logic makes sense only
if p^ < l, implying that cl must be smaller than l: Symmetric argument implies that ch must be
above h: In fact, we show that in the limit
! 1 so that aggregate shocks are unimportant, cl = l
and ch = h: As …rms understand that the price of capital in the idiosyncratic stage is p^ = c ;
when only that stage matters, they decide to (dis)invest exactly when that price reaches the cost
of (dis)investing.
Turning to the other parameters, the higher l and RK ; and the lower h and RC (i.e., the lower
the adjustment cost and the higher the relative bene…t of capital to cash), the less the cash bu¤er
that the planner is willing to build up. This reduces the upper threshold cPh , as stated in the second
result in Proposition 5.
These results immediately imply that the disinvestment threshold is too high in the market
equilibrium. Proposition 5 (or see Figure 4) suggests that the investment threshold ch in the
market equilibrium can be either higher or lower than cPh in the planner’s solution, depending on
the parameters. That is, our economy might feature underinvestment always, or underinvestment
during recessions but overinvestment during booms. In the next subsection, we identify the subset
of parameters for the latter case, call it a two-sided ine¢ ciency, and further explore the underlying
mechanism.
3.2
Two-sided ine¢ ciency
The following proposition states the main result of our paper.
Proposition 6 Under the conditions of Proposition 1, the following statements hold:
1. Firms dismantle capital before the cash-to-capital ratio reaches zero, i.e., cl > 0. Hence the
20
market equilibrium implies underinvestment in capital and over hoarding of cash in recessions.
2. If the di¤ erence between the productivity of capital and that of the new investment opportunity,
RK =h RC is su¢ ciently small, then we have ch < cPh . That is to say, the market equilibrium
implies overinvestment in capital during booms.
Figure 2 illustrates a case of two-sided ine¢ ciency. The thin, dashed curves on panels A-D of
Figure 2 illustrate the properties of the solution of the planner’s problem. Panels B and C show
the planner’s marginal valuation of cash and capital in the aggregate stage, while panel A shows
the ratio of the two, which is the shadow price of capital as de…ned in (24). The dashed (solid)
vertical lines show the thresholds of the market equilibrium (planner’s problem). As explained,
in the market equilibrium …rms dismantle capital when some cash is still around, cl > 0. In this
example, …rms create new capital at a lower liquidity level than the social planner would, ch < cPh :
Panel D contrasts the resulting evolution of cash-to-capital ratio in the planner’s solution and in
the market equilibrium under the same sample path of shocks fdZt g. Proposition 4 implies that,
in the case of two-sided ine¢ ciency, the resulting investment waves are too volatile in the market
equilibrium (illustrated by Panels E and F) in the sense that the expected adjustment intensity
of capital is too high.15 Finally, Proposition 3 implies that any policy that raises (decreases) the
upper investment (lower disinvestment) threshold would unambiguously increase total welfare in
this case.
The reason for the di¤erence between the planner and the market is a wedge between the
private and social valuation of capital relative to cash. To see this, consider …rms’ marginal rate
of substitution (MRS) between capital and cash in the idiosyncratic stage. As the value of capital
is RK and p^ RC and the value of cash is
RK
p^
the MRS is
M RS
and RC for capital …rms and cash …rms respectively,
1
p RC + RK )
2 (^
RK
1
2 RC + p^
= p^ :
(26)
Intuitively, when capital price, p^ ; is higher, …rms value more the capital they own, because cash
…rms can sell their capital and capital …rms must buy the capital they lack at that higher price. In
contrast, the relative social value of capital to cash is always
RK
RC .
From the social perspective, the
main function of the idiosyncratic stage is that it allocates cash and capital to the highest-value
user. The corresponding transfer across the two type of …rms in the market equilibrium, pinned
down by p^ ; is immaterial for the planner!
The wedge between the social and private valuation in the aggregate stage naturally follows from
the wedge in the idiosyncratic stage. Because the price in the decentralized economy guides each
individual …rm’s investment decisions, it is this wedge that drives the ine¢ ciency of the investment
waves in the aggregate stage. What is more, in our model the valuation wedge ‡uctuates with the
aggregate liquidity state. When
is …nite, because …rms are worried about cash-‡ow shocks, the
15
When comparing Panels E and F, recall Proposition 4 and the de…nition of T (c; ch ; cl ). This excess volatility is
in terms of the expected total adjustment intensity. Note that we cannot say whether, conditional on investment,
the size of the adjustment is larger or smaller in the market equilibrium than in the planner’s solution.
21
aggregate liquidity holding of …rms is procyclical. Our contracting frictions imply that p^ positively
depends on aggregate liquidity holdings. Therefore, from (26), the private incentive to hold capital
instead of cash decreases during recessions with low prices and increases during booms with higher
prices, compared with its social counterpart.
Figure 2 shows our mechanism in action. In Panel A, the solid line shows the market value of
capital relative to cash in the aggregate stage, and the dashed line shows its social counterpart.
The di¤erence between them comes from our wedge.
Although underinvestment in recession is independent of the parameters, whether there is overor underinvestment during booms depends on the parameter values. For example, consider again
the case
! 1: Recall that RK > hRC implies cPh = 0 in this limit, while, as p^ drives investment
and disinvestment in the decentralized case, ch = h and cl = l: That is, we have underinvestment
both during booms and recessions: To generate overinvestment during booms, we need to make
capital less attractive relative to cash for the planner.
As Proposition 5 describes, starting from a very large ; decreasing
does exactly that. In fact, as Proposition 6 shows, by decreasing RK
zero to any positive level within [0; RK ]. While a smaller RK
and/or increasing RC
hRC ; we can raise cPh from
hRC makes cash more attractive in
both the market and planner’s solution, its e¤ect on the market solution is much smaller because
of the additional private incentive force we described above. Loosely speaking, this additional force
keeps ch and cl close to h and l in the market equilibrium. Therefore, as Proposition 6 states, when
hRC is smaller than a given threshold, cPh > ch i.e., we have overinvestment during booms.
RK
We conclude this analysis with three remarks.
Remark 1 We can push the foregoing point further. So far, our analysis is performed under the
parameter restriction of
RK
RC
> h. What if l <
RK
RC
< h, which says that capital is more attractive
given the relatively small liquidation bene…t and that cash is more attractive given the relatively large
capital building cost? In Additional Material we show that, in this case, we always have a two-sided
ine¢ ciency. The intuition of the limiting case
! 1, which corresponds to a two-period static
model, is rather simple. Given the relatively high adjustment cost, the planner would never want to
convert capital to cash or vice versa, implying that cPl = 0 and cPh = 1. But in the market solution
with cl = l; ch = h remains the same as in our base setting, as …rms still make their (dis)investment
decisions in response to the market price p^ = c . It follows that 0 = cPl < cl < ch < cPh = 1,
hence the two-sided ine¢ ciency.
Remark 2 We emphasize that our market ine¢ ciency result comes from non-contractible idiosyncratic investment opportunities. Without contracting frictions, say if RK and RC were pledgeable,
p^ =
RK
RC
would always hold and there would be no wedge between the private and social relative
value of capital to cash. Just as in our baseline model, in the absence of contracting frictions …rms
still build up cash bu¤ ers against negative cash-‡ow shocks, because the precautionary motive to hold
cash is still present. However, in this variant the investment and disinvestment thresholds are the
same in both the planner’s solution and the market equilibrium (for a formal proof, see Additional
22
Material). That is, the precautionary motive alone does not create ine¢ ciency. Nevertheless, in
our model this precautionary motive to hold cash interacts with the externality.
Remark 3 Our mechanism is closely related to the main intuition behind the seminal papers on
welfare a¤ ecting pecuniary externalities of Stiglitz (1982), Geanakoplos and Polemarchakis (1985),
and Greenwald and Stiglitz (1986), which are followed by the more recent work of Shleifer and
Vishny (1992), Allen and Gale (1994, 2004, 2005), Caballero and Krishnamurthy (2001, 2003),
Lorenzoni (2008), Farhi, Golosov and Tsyvinski (2009), Farhi and Tirole (2012) and Gale and
Yorulmazer (2011). The critical observation in these papers is that, because of frictions, agents’
marginal utility of wealth might not be equalized across time or states in the decentralized equilibrium. In this case, a price change can work as a transfer from low marginal utility states to high
marginal utility states, creating ex ante welfare improvement. Indeed, there is a parallel argument
in our model, as the marginal utility of wealth in the idiosyncratic stage is
RC for cash …rms. Whenever p^ <
RK
RC ,
RK
p^
for capital …rms and
then the marginal value of wealth is higher for a capital
…rm. Therefore, if an intervention were to push down the threshold cl to cl
" at that state, the
delayed disinvestment would lower the capital price at the idiosyncratic stage pb ; which would be a
transfer from the cash …rms (sellers of capital) to capital …rms (buyers of capital), increasing ex
ante utility. However, note that, unlike in many other models in this literature, in our case it is
not the transfer per se that is the source of the welfare e¤ ect, but the more e¢ cient investment in
the aggregate stage.16
4
Applications
In the …rst part of this section, as a main application, we discuss the role and limitations of economic
policies in our context. In the second part we o¤er further applications connecting our …ndings to
sectoral investment cycles and …nancial development.
4.1
Economic policies
Proposition 3 shows how social welfare in our economy changes as the investment and disinvestment
thresholds change. However, in a market economy the policymaker cannot set these thresholds directly. Instead, the policymaker might be able to in‡uence the investment/disinvestment threshold
by changing the relative incentives of holding cash and capital, that is, by a¤ecting the market
price. In this section, we are interested in how various types of economic policies can serve this
purpose. We …rst make the following de…nition:
16
To see this, …x the aggregate stage investment but consider an unexpected intervention in the beginning of the
idiosyncratic stage. This unexpected intervention transfers " cash from cash …rms to capital …rms and then allows
them to trade, produce, and consume. As cash …rms would still exchange all their capital for cash and vice versa, the
ex post allocation of the given (K; C) across …rms would remain the same, hence this intervention would not a¤ect
ex ante welfare. A similar intervention in the bad state of the interim period of Lorenzoni (2008), i.e., transferring
cash from consumers to …rms, would change the amount of disinvestment, hence a¤ecting welfare.
23
De…nition 2 A balanced (budget-neutral) policy is an intervention that changes the marginal value
of capital only by the transfer scheme
for each unit of capital held, and
(c) ? 0, such that, given c, c (c) is the e¤ ective transfer
(c) is the e¤ ective transfer for each unit of cash held. An
intervention equilibrium is a market equilibrium where the wealth dynamics in (4) is adjusted by
transfers.
We refer to the equilibrium objects in an intervention equilibrium by the superscript : In an
intervention equilibrium, the policymaker in‡uences the outcome only through the e¤ect of
(c)
on the price in the aggregate stage.
The family of balanced policies is rich, because
(c) might be de…ned and implemented in
various ways. The simplest case is to impose a particular transfer between cash holders and capital
holders. But the policymaker, to avoid ine¢ cient liquidation, might also target a certain price path,
p (c) ; which will implicitly de…ne
might implement
(c) : If
(c) is positive in some range of c; the policymaker
(c) by buying a fraction of capital above market prices and selling it back to the
market at some point. That is, in our abstract world, it is immaterial to the welfare e¤ect whether
policymakers choose to provide subsidies or bailouts to certain industries, to impose measures which
a¤ect the collateral value of assets, or to implement asset purchase programs, as long as the implied
marginal transfers
(c) in these programs are the same.
Note that by the argument derived in Section 3, the (scaled) value of the representative …rm
in an intervention equilibrium is still jP (c; cl ; ch ) as de…ned in (20), where cl ; ch are the implied
investment/disinvestment thresholds. Therefore, Proposition 3 continues to hold: the welfare e¤ect
of a policy can be traced back to its e¤ect on the thresholds.
In the rest of this section, we analyze interventions concentrated on certain stages of our investment waves. Since the 2008 …nancial crisis, there has been an ongoing debate on the potential
adverse e¤ects of interventions during recessions on incentives during booms and, relatedly, on the
optimal mix of ex ante regulation and ex post intervention (e.g., Diamond and Rajan, 2011; Farhi
and Tirole, 2012; Jeanne and Korinek, 2013). Our modelling approach emphasizes that a policy
that, say, makes capital more attractive in a recession, a¤ects the relative value of capital in every
other state. What is more, the e¤ect of that policy on the investment threshold in booms feeds back
to agents’welfare in recessions, too. As we demonstrate, when a two-sided externality is present,
this interaction adds an interesting layer to this discussion.
We start our analysis with the following de…nition.
De…nition 3 A balanced policy is concentrated on low (high) states if
(c) = 0 for any c > c0
(c < c0 ):
The next proposition speci…es a simple criterion to decide how such a concentrated policy a¤ects
welfare.
Proposition 7 The following statements hold.
24
1. An intervention concentrated on low states to decrease the disinvestment threshold cl < cl ,
also reduces the investment threshold ch < ch ; if p (c0 ) > p (c0 ) and q (c0 )
increases the investment threshold ch > ch ; if p (c0 ) < p (c0 ) and q (c0 )
q (c0 ) : It
q (c0 ) :
2. An intervention concentrated on high states to increase the investment threshold ch > ch ,
also increases the disinvestment threshold cl > cl ; if p (c0 ) < p (c0 ) and q (c0 )
decreases the disinvestment threshold cl < cl ; if p (c0 ) > p (c0 ) and q (c0 )
q (c0 ) : It
q (c0 ) :
Proposition 7 states that, to understand a policy’s welfare consequences, it is su¢ cient to check
the e¤ect of the policy at the single state c0 , where the intervention stops. Together with Proposition
3 it also provides clear guidelines to the policymaker. For example, suppose that the economy
features two-sided ine¢ ciency. The policymaker might want to avoid ine¢ cient liquidation by
implementing a policy that increases the price of capital and decreases the value of cash in recessions.
As long as the policy has the same e¤ect at c0 ; it unambiguously worsens the overinvestment problem
during the boom. However, if the policymaker manages to …nd an alternative that partially avoids
ine¢ cient liquidation and decreases the price of capital (without decreasing the value of cash) at
c0 at the same time, then the intervention improves welfare everywhere.
To illustrate the usefulness of these guidelines, consider a particular family of policies. There,
the policymaker chooses a price ‡oor for capital l + with
0 together with a lower disinvestment
threshold cl with cl < cl ; and designs a policy that does not allow the price to fall below l +
as long as c
cl : With this intervention, the policymaker ensures that capital is liquidated only
at cl . As we show in Online Appendix B.4, the choice of
the corresponding transfer scheme
implement the transfer
and cl endogenously determines
(c) and the intervention threshold c0 : The policymaker can
(c) as a direct subsidy to capital holders, or, for example, initiates a
tax-…nanced program of buying assets at a markup
state-dependent intensity
(c),
above the market price p (c) with some
where17
(c) =
(c) ((p (c) + ) q (c)
v (c)) :
The dashed and dotted curves in Figure 5 show the main equilibrium objects of the intervention
equilibrium for a
intervention with
= 0 and a
> 0 price-‡oor policy with the same liquidation thresholds cl . The
= 0 slightly decreases the price and increases the upper investment threshold
ch , thereby increasing the welfare everywhere. In contrast, the intervention with
opposite e¤ect. The following proposition states the general result for the
> 0 has the
= 0 case when
is
large.
Proposition 8 A price-‡oor policy with
c 2 [cl ; ch ] as long as
= 0 and any cl < cl improves welfare at every state
is su¢ ciently large.
17
The policymaker can resell the purchased assets to …rms at the market price immediately, or after holding them
for a given time interval. The latter case might be a reasonable description of the TARP program implemented by
the US government in 2008.
25
Figure 5: Panels A-C depict the price of capital, the value of cash, and the value of capital in the baseline case (solid),
under price ‡oor policies = 0:05l (dotted) and = 0 (dashed). (In Panel A, the solid and dashed curves are on top
of each other.) Panel D depicts the percentage change in the social value due to the intervention. The vertical lines
in each panel from left to right are the liquidation threshold cl of both interventions, the intervention thresholds c0
of = 0 and of = 0:05l, and the investment thresholds of = 0:05l; the baseline case, and = 0: The horizontal
lines in Panel A are at the levels of l and h: Parameter values are RK = 4:1; 2 = 1; = 0:1; RC = 2; l = 1:8, h = 2,
and cl = 0:85:
The intuition behind the opposite e¤ect of the two interventions is instructive. The high price
‡oor close to the recession increases the value of capital during booms, encouraging more overinvestment. While there is less ine¢ cient liquidation during the recession due to the support for
capital holders, …rms— even during the recession— foresee the resulting stronger overinvestment
during booms. In the numerical example in Figure 5 with
> 0, the latter e¤ect dominates the
earlier and decreases welfare.
In contrast, when the price ‡oor is su¢ ciently low, the policymaker prolongs the near-recession
state of the economy by keeping the price very close to its minimal value through states of mild
recovery. This can decrease the relative value of capital when the economy is booming. Thus, even
if the policymaker manages to avoid some ine¢ cient liquidation, the intervention can still decrease
the value of capital, which makes the overinvestment problem less severe.
One take-away from this experiment is that setting the appropriate price is critical. If the set
price for the recession is not su¢ ciently low, economic agents foresee the induced overinvestment
in booms, thus decreasing welfare during both booms and recessions. If the set price is too low,
then it does not stop ine¢ cient liquidation. Hence, the policymaker should set a price ‡oor that
just discourages …rms from selling the assets for lower-user value agents. This policy endogenously
26
keeps the price of capital low through mild states of recovery, which helps curb incentives for
overinvestment during booms.
4.2
Sectoral and aggregate investment waves
In this section we ‡esh out two further applications that relate our model to sectoral investment
waves and …nancial development. We keep the discussion here brief, and expand on these applications and on the related literature in He and Kondor (2012).
Sectoral investment waves It is well known that certain industries go through boom-and-bust
patterns. Hoberg and Phillips (2010) argue that these patterns are widespread in the data, well
beyond the handful of well-known episodes such as the 1990s tech-bubble and the 1980s biotech
bubble. Interpreting our economy as one sector, our model implies that such cycles arise naturally,
even if the technology does not change. The main implication of our mechanism is that in sectors
with more non-contractible investment opportunities (e.g., sectors with a larger share of intangible
input), other things being equal, these cycles are less e¢ cient. That is, in these sectors too many
resources would be spent on frequent adjustment of the capital level, reducing pro…tability.
Financial development and investment dynamics Our model also suggests a novel rationale
for stylized facts on the connection of …nancial development and investment dynamics. Aghion et al.
(2010) provide a useful starting point. The authors decompose aggregate investment to structural
and other investment, arguing that structural investment is a proxy for investment in longer-term,
riskier, but more productive projects. Then they show that in less …nancially developed countries
structural investment is much more sensitive to productivity shocks, implying a more volatile and
more procyclical pattern. They suggest that this di¤erence in the dynamics of the composition
of investment activities is an important way in which the lack of …nancial development hinders
growth.
Our results are broadly consistent with the stylized facts in Aghion et al. (2010), if we take
capital as a proxy for more productive and riskier projects, and the lack of contractibility on
future investment opportunities as a proxy for a low level of …nancial development. Our two-sided
ine¢ ciency implies more volatile investment in capital (Proposition 4), a lower level of expected
consumption (Proposition 3), and a lower growth rate of the economy in the long term for less
…nancially developed countries.
5
Robustness
In this section, we present variants of our baseline model to show that the presence of a two-sided
ine¢ ciency is not linked to particular technical features of our model. First, we present a variant
where the aggregate and idiosyncratic stages are not separated. We show that, due to a similar
27
intuition in the baseline model, two-sided externalities are present. Second, we show that allowing
for collateralized borrowing could make the presence of two-sided externalities even more prevalent.
5.1
Contemporaneous aggregate and idiosyncratic shocks and a single market
We argue in Section 2.2 that the separation of the market into two segments, that in which …rms in
the aggregate stage trade and that in which …rms in the idiosyncratic stage trade, is only a technical
device. In this part, we build a variant in which we eliminate this segmentation of markets.
Just as in the alternative interpretation of our baseline case in Section 2.4, here we think of
…rms facing i.i.d. chances of being hit by idiosyncratic investment opportunities occurring with
intensity . Applying the law of large numbers, every instant there are a dt fraction of …rms hit
by the idiosyncratic skill shocks. These …rms face the same investment opportunity sets as in our
baseline model: with half probability each …rm becomes either a capital or a cash …rm. Again, as
in our baseline, before …nal production …rms can trade their holdings. However, in this variant,
both …rms with the investment opportunity and those without it trade at the same Walrasian
capital price pt . That is, there is no separate price p^ for the idiosyncratic stage. Firms with
investment opportunity, after exploiting it, exit and consume, but the economy goes on forever
with the remaining …rms who have not got any investment opportunity yet.
As the baseline model, incumbent …rms face aggregate cash-‡ow shocks according to (1), and
they can transfer cash to capital or vice versa at the same linear investment technology as in our
base model. We further assume that in the aggregate stage there is another sector that combines
capital and cash to produce perishable …nal consumption goods at the Cobb-Douglas technology
K C1
, where
2 (0; 1) and
> 0 are positive constants. As we will discuss shortly, the
Cobb-Douglas technology with Inada condition is only for ease of illustrating the welfare e¤ect of
small policy interventions.
We can solve the model by keeping track of the same state variable, cash-to-capital ratio ct =
Ct =Kt . There will be an upper (lower) boundary ch (cl ) so that …rms start investing in (liquidating)
capital when ct hits the boundary from below (above). Inside the inaction region ct 2 (cl ; ch ), given
the endogenous capital price p (ct ), the cash-to-capital ratio follows
dct =
(p (ct ) + ct ) dt +
| 2
{z
}
cash …rms exiting w. cash
|
ct
2
ct
1+
p (ct )
{z
dt
}
+ dZt =
2
p (ct ) +
2
dt+ dZt : (27)
capital …rms exiting w. capital
Here, we have labeled the extra drift terms relative to (6). For example,
…rms causes an out‡ow of
c2t
p (ct )
1+
translates to a positive drift of
ct
2
ct
p(ct )
1+
2 dt
fraction of capital
dt on the scaled (by capital Kt ) aggregate capital, which
ct
p(ct )
dt for the cash-to-capital ratio.
Although the closed-form solution is no longer available once the drift of the state variable
depends on the endogenous capital price p (c), we can study the market equilibrium by numerically
28
solving a system of ODEs. For the marginal value of cash q (c), we have
0 = q 0 (c)
2
|
2
c2
p (c) +
+ q 00 (c) +
p (c)
2
|
{z
}
1
2
e¤ect of drift
RC +
RK
p (c)
{z
idiosyncratic stage
q (c) + (1
)c :
|
{z
}
} aggregate stage
(28)
We highlight three terms in (28). The …rst term captures the drift of the state variable ct as …rms
are exiting the economy. The second term gives the marginal value of cash when hit by idiosyncratic
shocks: Each unit of cash either yields RC if the …rm becomes cash type, or RK =p (c) if capital
type, each with half probabilities.
The third term,
(1
)c
=@
K C1
=@C, which is new, gives the marginal value of
cash at the aggregate stage before being hit by idiosyncratic investment opportunities. Due to
Inada condition of the Cobb-Douglas technology, this marginal bene…t
(1
)c
soars when c
falls towards zero, guaranteeing that in equilibrium …rms start liquidating capital for cash before c
hits 0. As a result, the equilibrium disinvestment threshold cl > 0 always takes an interior solution,
which helps us illustrate the e¤ect of a small distortionary tax scheme that we consider later. For
details, see Additional Materials.
Similarly the marginal value of cash v (c) satis…es
0 = q 0 (c)
2
+v 0 (c)
2
|
2
c2
p (c) +
+ v 00 (c)+
p (c)
2
|
{z
}
e¤ect of drift
RC p (c) + RK
2
{z
v (c) +
: (29)
c1
| {z }
} aggregate stage
idiosyncratic stage
Finally, to pin down the market equilibrium, we have the same boundary conditions as in the base
model
v 0 (ch ) = q 0 (ch ) = q 0 (cl ) = v 0 (cl ) = 0;
(30)
and (dis)investment optimality conditions are
v (ch )
v (cl )
= h; and
= l:
q ch
q cl
(31)
For the planner’s solution, we consider a policy from the family of balanced tax/subsidy considered in Section 4.1:
c (c) =
where
h,
l,
h
and
l
8
>
<
>
:
h
l
if c > (1
h ) ch
if c < (1 +
l ) cl
0
(32)
otherwise
are positive constants. This policy taxes capital during booms (speci…-
cally, when the economy is close to investment boundary ch , or, a
h
fraction below ch ) and/or
subsidizes capital during recessions (speci…cally, when the economy is close to disinvestment boundary cl , or, a
l
fraction above cl ). Budget-neutral policies imply that cash receives transfers of
(c). We obtain the new market equilibrium with intervention by numerically solving the new in-
29
vestment/disinvestment thresholds cgl ; cgh , joint with new marginal capital and cash values v g ( )
and q g ( ), respectively.18 As before, the (scaled) social welfare in the economy is measured as
j g (c) = v g (c) + cq g (c).
5.1.1
Two-sided ine¢ ciency
As an illustration, the solid curves in Panels A, B, and C on Figure 6 show the price of capital,
the value of capital, and the value of cash, all as a function of c in the alternative setting. We
observe that, in the alternative setting without segmented markets, the equilibrium objects are
qualitatively similar to those in the base model. The dashed curves on the same panels show the
corresponding objects under the policy intervention speci…ed in (32). We observe that cgl < cl and
cgh > ch . The intuition is clear: As individual …rms adjust their real decisions based on market
prices, taxing (subsidizing) capital during booms (recessions) postpones the market investment
(disinvestment) during that time. Because the policy intervention is small (we set
and
h
=
l
h
=
l
= 1:5%
= 7% in this numerical example), the quantitative e¤ect of this policy in Panels A, B,
and C is only slightly visible.
Panel D shows the resulting improvement in social surplus for this policy (“two-sided intervention,” solid, U-shaped curve). To show the two-sided ine¢ ciency more convincingly, we also
calculate the change in social welfare under two di¤erent one-sided intervention policies, i.e., either
only taxing capital during booms (“upper intervention” sets
l
= 0 , dotted, increasing curve) or
only subsidizing capital during recessions (“lower intervention” sets
h
= 0, dashed, decreasing
curve). Each policy generates a strictly positive value improvement, and the curve of the two-sided
intervention is the upper envelope of the one-sided interventions. These results imply the twosided externality with underinvestment during recessions and overinvestment during booms in this
economy.
For this example, we chose parameters satisfying l <
RK
RC
< h and picked a large
. Then,
the intuition for two-sided externality follows from the argument in Remark 1 in Section 3.2. In
particular, the parameters imply that the …rm’s private relative valuation of capital to cash, i.e.,
its private M RS (which is close to pt 2 [l; h] according to the arguments in Section 3.2) will vary
around its social counterpart (which is close to
RK
RC
according to the arguments in Section 3.2).
Hence, in booms, when pt is close to h; the private valuation of capital is higher than its social
counterpart. This implies that the investment threshold is lower (so the …rm invests earlier) in the
market equilibrium. The opposite argument holds during recessions, implying two-sided ine¢ ciency.
5.2
Collateralized borrowing
We now go back to the base model and introduce collateralized borrowing. We present here only
a sketch of this extension and the main results. A detailed analysis can be found in Additional
Material, which is available on the the author’s website.
We have the same boundary conditions p (cgl ) = l and p (cgh ) = h, but modify the ODEs slightly to re‡ect transfers:
For capital, we need to add c (c) in (28); for cash we need to subtract (c) in (29).
18
30
Figure 6: Panels A-C depict the price of capital, the value of cash, and the value of capital in the alternative setting
both in the market equilibrium (solid) and when the planner decreases the lower threshold and increases the upper
threshold (dashed). Panel D depicts the percentage change in value due the intervention. The vertical lines in each
panel from left to right are the disinvestment threshold of the intervention and in the market equilibrium, cgl = 0:6457,
cl = 0:7857; and the investment threshold in the market equilibrium and of the intervention, ch = 3:5729; cgh = 3:7620:
Parameter values are RK = 4:2; 2 = 1; = 5; RC = 2; l = 1:7, h = 2:6, = 0:5, h = l = 7%, and h = l = 1:5%.
We use MATLAB built-in ODE solver bvp4c to solve the model, with a convergence criterion of 10 7 .
We assume that installing one unit of capital allows the …rm to borrow a constant b 2 [0; l) units
of cash from external creditors, during both the aggregate and the idiosyncratic stages, and there
are no other borrowing technologies allowed.19 The welfare accounting remains the same, because
external creditors obtain zero rent always. We can characterize both the market equilibrium and
the planner’s solution in closed form for the economy with collateralized borrowing, using the fact
that, in equilibrium, …rms always maximize out their borrowing capacity.
Our main observation is that for a set of economies with small h l, in the absence of borrowing
…rms underinvest even during booms, but, once allowing for collateralized borrowing, …rms will start
overinvesting during booms. This result indicates the potential social cost of excessive collateralized
borrowing. Formally, de…ne "
h
l, and imagine improved borrowing technologies modeled as
an increasing sequence b (k) > 0 for k 2 (0; 1), so that b (k)
l
O "k : The higher the k, the
smaller the distance O "k , hence the higher the borrowing capacity b (k) …xing l.
Proposition 9 We have the following results:
19
For example, …rms cannot borrow against the new investment opportunities at the idiosyncratic stage, potentially
because of the non-contractible nature of new opportunities. Here, b can be interpreted as the ine¢ cient recovery
that external creditors can obtain if the borrower reneges. This is in the tradition of Kiyotaki and Moore (1997), but
to avoid complication we do not link the borrowing capacity to endogenous capital prices.
31
1. When " ! 0; in the market equilibrium both ch and cl converges to l. However, in the social
planner’s solution we have cPl = 0 and cPh ! 0: Hence, there is underinvestment during both
booms and recessions.
2. For su¢ ciently small ", there is overinvestment during the boom ch (k) < cPh (k) for k > 1=3.
For the …rst part, from the planner’s view, cash-‡ow shocks pose little risk given little adjustment
cost (as h is close to l). Since capital is more productive (RK > hRC ), the planner should hold
almost no cash (i.e., cPh ! 0). However, the wedge between social and private incentives of holding
capital remains. If ch ! h, cl ! l, and h ! l; then the price in the idiosyncratic stage is close to l;
which suggests that in the competitive market …rms liquidate capital as ct falls below l and build
as it rises above l:
The second result of Proposition 9 gives the main point of this subsection. A higher k implies
a higher b (k), which increases the value of capital for both the planner and the market. We show
that the positive e¤ect on the market is stronger and, as b (k) gets close to l, at some point ch < cPh ,
i.e., overinvestment during booms. Intuitively, the planner’s solution is mainly determined by the
adjustment cost h
l = ". For the market, as k increases, a higher b (k) = l
O "k drives up l and
h = l + ", which raises the capital price in the idiosyncratic stage and hence also strengthens the
private incentives to hold capital in the aggregate stage. Therefore the increase of the borrowing
capacity leads to a faster decrease in ch than in cPh :
6
Conclusion
We build an analytically tractable, dynamic stochastic model of investment and savings, in which
investment cycles, i.e., boom periods with abundant investment and bust periods with low investment, arise naturally. In the presence of non-contractible idiosyncratic investment opportunities,
a two-sided ine¢ ciency can arise: there is too much investment in risky capital and cash bu¤ers
are too low during booms, and there is too little investment and too much cash hoarding during
recessions. We show that in this case a one-sided policy targeting only the underinvestment during
downturns might be ex ante Pareto inferior to the absence of intervention in all states (including
downturns).
We acknowledge that there are standard ways to eliminate the ine¢ ciency studied in this paper.
In the working paper of this article (He and Kondor (2012)), we investigate this question in a
simpli…ed two-period setting. From an ex ante perspective, we show that the market can be
completed by introducing Arrow-Debrew securities contingent on the realization of idiosyncratic
opportunities, which restore investment incentives of individual …rms in the competitive market.
However, if idiosyncratic investment opportunities are not veri…able, so that the enforcement of
Arrow-Debrew securities requires self-reporting, in general private investment incentives are still
distorted away from the social ones. From an ex post perspective, if the …nal production output is
fully pledgeable, then there will be no wedge between the social and private value of capital to cash,
32
even without Arrow-Debrew securities.20 Nevertheless, this result breaks down once we introduce
imperfect delegation, which can be further microfounded by lack of commitment, hidden e¤ort,
or even information processing. In sum, just as in our two-period setting the social value ratio is
independent of the state of cash-to-capital, our key result holds as long as market imperfections
lead the price of capital to increase with the cash-to-capital ratio (in our model the idiosyncratic
stage price p^ = c takes its extreme form due to the cash-in-the-market pricing).
Apart from analyzing two-sided ine¢ ciencies, we also presented a novel way of modeling dynamic
investment and savings. This method provides analytical tractability in a dynamic stochastic
framework for the full joint distribution of states and equilibrium objects. Exploring the potential
of the developed framework in various other contexts is a task for future research.
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A
Appendix
In this Appendix we provide proofs for Propositions 2, 3, 4, the …rst part of Proposition 5 and Proposition
6. The proof of Proposition 9 is given in Additional Material available on the author’s website. Proofs for
all the other results are provided in Online Appendix.
35
A.1
Proof of Proposition 2
Based on boundary conditions RK + D1 + D2 = l (RC
solutions for D1 and D2 are given by
P
RK
(RK lRC ) e2 ch
, D2 =
P
(1 + l ) e2 ch (1 l )
(1 + l ) e2
D1 =
cP
h
D1 + D2 ) and D1 e
cP
h
lRC
(1
l )
+ D2 e
cP
h
:
= 0, the
(A.1)
00
To verify that cP
l = 0; we need to show that jP (0) < 0:
1
j 00 (0) = D1 + D2 =
2 P
(RK
lRC )
e2
P
e2 ch
cP
h
1
P
e2 ch
+l
< 0:
+1
1
The super-contact condition at the optimal upper investment threshold cP
h is
P
@ 2 jP c; cP
l ; ch
@c
0=
=
2
D1 e
cP
h
+ D2 e
cP
h
:
(A.2)
c=cP
h
To show cP
h exists and unique, de…ne a function G (c) (G (c) is proportional to (A.2) if we plug D1 and D2
in (A.1) into (A.2))
G (c)
with G (0) = 2RK
l h
RK lRC
RK
RK
hRC c
e (1 + l )
lRC
< 0 (recall RK
G0 (c) =
(RK
(RK
hRC > RK
(1
l )e
c
2 (c + h) ;
(A.3)
lRC > 0) and G (1) = 1. We have
hRC )
(l + 1) ec + e
lRC )
c
(1
l )
2 ;
G0 (0) = 2RC RKl hlRC < 0, and G0 (c) changes sign only once. Consequently, there is a unique c^ that
G0 (^
c) = 0, implying that G (c) is decreasing for c < c^ and increasing for c > c^: As G (0) < 0 and G (1) = 1;
P
there must be a unique cP
h that G ch = 0, verifying the equation (23).
The social planner’s value function jP (c) satis…es
2
0=
2
jP00 (c) + (RK + RC c
jP (c))
(A.4)
0
P
00
P
with boundary conditions jP (0) = ljP0 (0) ; jP cP
= h + cP
= 0: Note that the
h
h jP ch ; and jP ch
P
P
boundary conditions imply that jP ch = RK + RC ch . For later reference, we show that jP (c) is concave
P
and increasing over 0; cP
h ; and jP (c) < RK + RC c for c 2 0; ch First, from smooth pasting condition at
P
ch we have
jP cP
RK + RC cP
RC h RK
h
h
RC jP0 cP
=
R
=
< 0:
C
h = RC
P
P
h + ch
h + ch
h + cP
h
Then, taking derivative again on (A.4) and evaluate at the optimal policy point cP
h , we have
jP000 cP
h =
and as a result jP00 cP
h
2
2
RC
jP0 cP
h
=
2 RK RC h
> 0;
2
h + cP
h
< 0. Suppose that jP fails to be globally concave over 0; cP
h . Then there exists
36
00
some point jP00 > 0, and pick the largest one b
c so that jP00 is concave over b
c; cP
c) = 0 and jP000 (b
c) < 0.
h with jP (b
2
00
P
0
0
000
0
P
But since jP is concave over b
c; ch , jP (b
c) > jP ch > RC , therefore 2 jP (b
c) = (jP (b
c) RC ) > 0;
contradiction. Therefore jP is globally concave over 0; cP
,
which
also
implies
that
j
(c)
<
RK + RC c for
P
h
c 2 0; cP
due
to
(A.4).
h
A.2
Proof of Proposition 3
P
Suppose that we are given the policy pair (cl ; ch ) with 0 < cl < ch < cP
h where ch satis…es the super-contact
00
P
P
condition jP ch ; 0; ch = 0: To avoid cumbersome notation we denote the social value jP (c; cl ; ch ) given the
policy pair (cl ; ch ) by j (c; cl ; ch ), and denote the social value under the optimal policy jP c; 0; cP
h by jP (c).
We need to show that
@j (c; cl ; ch )
@jP (c; cl ; ch )
< 0 and
> 0.
@cl
@ch
1 1
2 2
This result further implies that for 0 < c2l < c1l < c1h < c2h < cP
h , we have j c; cl ; ch < j c; cl ; ch :
00
00
As preparation, we …rst show that j (ch ; cl ; ch ) < 0 and j (cl ; cl ; ch ) < 0: Because (cl ; ch ) is sub2
optimal, we must have j (c; cl ; ch ) < jP (c)
RK + RC c (recall Proposition 2). Then 0 = 2 j 00 (c) +
(RK + RC c j (c)) implies that j (c) is strictly concave at both ends. Second, for any policy pair (cl ; ch )
(including the market solution or the planner’s solution), the smooth pasting condition (not optimality
condition!) at the regulated ends implies that
j (ch ; cl ; ch ) (ch + h) j 0 (ch ; cl ; ch ) = 0;
j (cl ; cl ; ch ) (cl + l) j 0 (cl ; cl ; ch ) = 0:
(A.5)
(A.6)
Now we start proving the properties for the upper investment policy ch . De…ne Fh (c; cl ; ch ) @c@h j (c; cl ; ch ),
which is the marginal impact of changing the investment policy on the social value. Di¤erentiating the ODE
2
@
(A.7) by the policy ch , we have 2 @c@h j 00 (c; cl ; ch )
@ch j (c; cl ; ch ) = 0, or
2
2
Fh00 (c; cl ; ch )
Fh (c; cl ; ch ) = 0:
(A.7)
Moreover, take the total derivative with respect to ch on the equality (A.5), i.e., take derivative that a¤ects
both the policy ch and the state c = ch , we have
@
@ 0
j (ch ; cl ; ch ) + j 0 (ch ; cl ; ch ) = j 0 (ch ; cl ; ch ) + (ch + h)
j (ch ; cl ; ch ) + j 00 (ch ; cl ; ch )
@ch
@ch
@
@ 0
)
j (ch ; cl ; ch ) (ch + h)
j (ch ; cl ; ch ) = (ch + h) j 00 (ch ; cl ; ch ) < 0
@ch
@ch
) Fh (ch ; cl ; ch ) (ch + h) Fh0 (ch ; cl ; ch ) < 0:
(A.8)
which gives the boundary condition of Fh ( ) at ch . At cl we can take total derivative with respect to ch on
the equality (A.6), we have the boundary condition of Fh ( ) at cl :
@
@ 0
j (cl ; cl ; ch ) = (cl + l)
j (cl ; cl ; ch ) ) Fh (cl ; cl ; ch )
@ch
@ch
(cl + l) Fh0 (cl ; cl ; ch ) = 0:
(A.9)
With the aid of these two boundary conditions, the next lemma shows that Fh ( ) has to be positive always.
Because of the de…nition of Fh (c; cl ; ch ) @c@h j (c; cl ; ch ), it implies that raising ch given any state c and any
lower policy cl improves the social value. The argument for the e¤ect of cl is similar and thus omitted.
Lemma A.1 We have Fh (c) > 0 for c 2 [cl ; ch ].
Proof. We show this result in two steps.
37
1. Fh (c) cannot change sign over [cl ; ch ]. Suppose that Fh (cl ) > 0; then from (A.9) we know that
Fh0 (cl ) > 0. Then simple argument based on ODE (A.7) implies that Fh ( ) is convex and always
positive. Now suppose that Fh (cl ) < 0; then the similar argument implies that Fh is concave and
negative always. Finally, suppose that Fh (cl ) = 0 but Fh changes sign at some point. Without loss
of generality, there must exist some point b
c so that Fh0 (b
c) = 0, Fh (b
c) > 0 and Fh00 (b
c) < 0. But this
contradicts with the ODE (A.7).
2. De…ne Wh (c)
(l + c) Fh0 (c) so that
Fh (c)
Wh0 (c) =
2 (l + c)
(l + c) Fh00 (c) =
2
Fh (c) :
(A.10)
As a result, Wh0 (c) cannot change sign. Because we have Wh (cl ) = 0, Wh (c) = 0 cannot change sign
either.
3. Now suppose counterfactually that Fh (c) < 0 so that Wh0 (c) > 0. Then (A.10) in Step 2 implies that
1
(Fh (ch ) Wh (ch )) < 0. But we then have
Wh (c) > 0, and hence Fh0 (ch ) = l+c
Wh (ch ) = Fh (ch )
(l + ch ) Fh0 (ch ) = Fh (ch )
|
(h + ch ) Fh0 (ch ) + (h
{z
}
l) Fh0 (ch ) < 0;
(A .8), n egative
where we have used (A.8). This contradicts with Wh (c) > 0. Thus we have shown that Fh (c) > 0.
A.3
Proof of Proposition 4
2
1
The expected total investment activity T (c) solves 2 T 00 (c) = T (c) with boundary conditions T 0 (cl ) = l+c
l
1
and T 0 (ch ) = h+c
.
For
example,
at
c
=
c
,
a
positive
shock
hits
with
c
=
c
+
:
To
get
back
to
the
upper
h
h
h
K
; thus, we have
cash-to-capital ratio ch , the economy builds new capital of dK = h+c
h
T (ch + ) =
dK
1
+ T (ch ) =
+ T (ch ) , T 0 (ch ) =
:
K
h + ch
h + ch
Now we study the impact of policies ch and cl on T ( ; cl ; ch ). For illustration we analyze cl only; a similar
@
T (c; cl ; ch ); we have
argument applies to ch . De…ne F (c) @c
l
2
2
T 00 (c; cl ) = T (c; cl ) )
2
2
F 00 (c; cl ) = F (c; cl ) :
To determine boundaries for F , at ch we have T 0 (c = ch ; cl ) =
F 0 (c = ch ) =
On the other end, T 0 (c = cl ; cl ) =
1
l+cl
1
h+ch
which implies that
@ 0
T (c = ch ; cl ) = 0:
@cl
implies that F 0 (c = cl ) + T 00 (c = cl ; cl ) =
F 0 (c = cl ) =
(A.11)
1
2
(l + cl )
1
(l+cl )2
or
T 00 (c = cl ; cl ) < 0;
Here we used the fact that T 00 (c = cl ; cl ) > 0; this fact is implied by (A.11) together with T (c) > 0 by
de…nition.
38
Now we show that F (c) > 0 so that the total investment activity goes up for a higher cl . To see this,
…rst note that F (c) never changes sign. Otherwise, suppose that there exists some c1 so that F (c1 ) = 0.
If F 0 (c1 ) > 0 then it must be that F is convex and positive for c > c1 , which contradicts with F 0 (ch ) = 0.
Similarly we rule out F 0 (c1 ) < 0. If F 0 (c1 ) = 0, then combining with F (c1 ) we can solve for F (c) = 0 for
all c, contradicting with F 0 (cl ) < 0. Now since F (c) never changes sign, it su¢ ces to rule out F (c) < 0
always. If it were true, then F is concave always due to (A.11). This contradicts with F 0 (cl ) < 0 = F 0 (ch ).
As a result, F (c) > 0.
A.4
Proof of Proposition 5
Recall G (c) de…ned in (A.3). Since …xing c we have
lim
RK hRC
RK lRC
(ec (1 + l )
!1
(1
l )e
c
)
c
= 1;
to ensure that G cP
! 1 we must have cP
h = 0 as
h ! 0. This is the …rst part of the …rst statement.
For the second part of the …rst statement,
@G (c)
RK
=
@
RK
hRC
cec (1 + l ) + lec
lRC
c (l
1) e
c
+ le
c
2 (c + h) ;
which is positive for su¢ ciently large : Finally, from the proof of Proposition 2 we know that G0 cP
h > 0.
p
=@
@G
c
)
(
@cp
h
< 0 which concludes the …rst part. The second
Hence, for su¢ ciently large ; we have @ h =
G0 (cp
h)
RK hRC
part follows because RK lRC is increasing in RK and decreasing in RC , and
@G (c)
@h
=
@G (c)
@l
=
RC
ec (1 + l ) (1 l ) e c
2 < 0;
lRC
RC 1 e 2c + RK + RK e2( c )
> 0:
(RK hRC )
2
e c (RK lRC )
RK
2 (c + h) < 0 always. This implies that for limRK !hRC G cP
h =
RK hRC
c
c
e
(1+l
)
(1
l
)e
(
)
0 to hold, it must be that cph ! 1 so that limRK !hRC RK lRC
! 2 . This concludes
c
the …rst statement of the proposition. The second statement is in Online Appendix.
Finally, …xing any c we have limRK !hRC G (c) =
A.5
Proof of Proposition 6
The …rst statement comes from point 2 of Proposition 1. For the second statement, note that the proof
of Proposition 1 goes through without any modi…cation for the case when RK = RC h: That is, even
in the limit RC h ! RK ; the investment threshold in the market equilibrium ch is …nite, and under the
parameter restriction of Proposition 1 we have ch < RK . However, note that given any parameters, in the
limit RC h ! RK the solution to equation (23) diverges to 1. Due to continuity, we can …nd RK RC h
P
appropriately small so that cP
h su¢ ciently close to RK and hence ch < ch . And, even if the solution to
(23) is above RK , we can show that the resulting optimal investment threshold cP
h lies above RK and hence
ch < RK < cP
.
The
details
of
this
argument
are
in
the
Additional
Material
available
on the author’s website.
h
This completes the proof.
39
B
Online Appendix for He and Kondor (2015): Proofs and Derivations
In this Online Appendix we provide proofs for Lemma 1 and Proposition 1, the second part of Proposition
5, and Propositions 7, 8 and 9.
B.1
Proof of Lemma 1 and Proposition 1
We construct the proof in steps. In particular, we separate Proposition 1 into the following four Lemmas.
These four lemmas are su¢ cient to prove Proposition 1.
Lemma B.2 If the equation system (12)-(13), (7)-(9) has a solution where ch < RK , and both v (c) and
q (c) are increasing in the range c 2 [cl ; ch ], then Proposition 1 holds.
Lemma B.3 The system (12)-(13), (7)-(9) always has at least one solution.
Lemma B.4 If h
l is su¢ ciently small, then ch < RK :
Lemma B.5 q (c) is decreasing in c. If h
B.1.1
l is su¢ ciently small, then v (c) is increasing for c 2 [cl ; ch ] :
Step 1: Proof of Lemma 1 and Lemma B.2
Denote the dollar share of capital in the …rm’s asset holdings by it , so that it = Kti pt =wti . According to
our conjecture, the value function can be written as (recall the aggregate cash-to-capital ratio c = C=K)
J
Kt ; Ct ; Kti ; Cti
=
wti
"
1
i
t
q (ct ) +
i
t
pt
#
v (c) = J Kt ; Ct ; wti ;
is linear in wt . This is equivalent to J C; K; Kti ; Cti = Kti v (c) + Cti q (c) stated in the Lemma. Also, we
have the wealth dynamics, expressed in terms of capital share it , as
dwti =
d
i
t
dKti +
i i
t wt
1
(dpt + dZt ) :
pt
And, q (c) 1 has to hold as …rms can consume cash at the …nal date (and there is no discounting), which
implies d it = 0, i.e., …rms do not consume in the aggregate stage.
As the …rm is choosing capital share it , and the capital to build or dismantle dKti , the HamiltonianJacobi-Bellman (HJB) of problem (3) can be written as:
0 = max
d
i
i
d
t ;dKt
i
t
1
0
dKti + Jw;C Et [dwt dCt ] :
+ JC Et [dCt ] + JCC Et dCt2 + Jw Et (dwt ) + JK
2
The endogenous price dynamics (using Ito’s Lemma) is
dpt =
1
2
p (c) dt + p0 (c) dZt + dBtp
2 00
dUtp ;
where dBtp (dUtp ) re‡ects p at p (cl ) = l (p (ch ) = h). This is because in any market equilibrium …rms will
create (dismantle) capital if pt = h (pt = l), and keep doing it until the price adjusts. We derived the
boundary conditions in the main text. Also, by risk neutrality and the initial homogeneity of …rms, before
the …nal date the price of the capital has to make …rms indi¤erent whether to hold capital or cash. Otherwise
markets could not clear. We also explained that p^ = c .
40
Thus, inside the re‡ection boundary (cl ; ch ) the above HJB equation is (we drop i from now on)
9
8 2
1 2 00
p (ct )
0
0
00
2
=
<
+
q
(c
)
w
(
+
p
(c
)
)
w
q
(c
)
+
q
(c
)
w
t
t
t
t
t
t
t
t
t
c
2
pt
pt
i
h
0 = max
:
RK
1
t RK
t : + w
+ (1
+ 12 ptt RC ct + (1
q (ct ) ;
t 2
t ) ct
t ) RC
pt
Since the problem is linear in t ; in equilibrium …rms must be indi¤erent in their choice of t : Thus, we
can calculate the dynamics of the cash (capital) value by choosing t = 0 ( = 1). Setting t = 0 directly
implies (10). Choosing t = 1 gives
2
0=
2
q 00 (c) + q (c)
1
2
2 00
p (c)
+ q 0 (c)
p
1
( + p0 )
p
+
1
p
2
(RK + RC c)
q (c) p :
Since v (c) = p (c) q (c), v 0 = q 0 p + p0 q, and v 00 = q 00 p + 2p0 q 0 + p00 q, we can rewrite the above equation as (11).
Given that the ODEs for v (c) and q (c) were derived by substituting in t = 1 and t = 0; it is easy to see
that these functions can be interpreted as the value of a capital and that of a unit of cash. This implies that
J
C; K; wti
=
wti
1
i
t
q (c) +
i
t
pt
!
wti v (c)
= q (c) wt
verifying both Lemma 1 and our conjecture on the form of J C; K; wti .
B.1.2
Step 2: Proof of Lemma B.3
First, note that for any arbitrary ch and cl from (9), we can express A1 -A4 in (12)-(13) as functions of ch and
cl only. Substituting back to (12)-(13) we get our functions parameterized by ch and cl which we denote as
v (c; cl ; ch ) and q (c; cl ; ch ) : Evaluating these functions at c = cl and c = ch ; we get the following expressions.
De…ne
fl (cl ; ch )
gl (cl ; ch )
fh (cl ; ch )
gh (cl ; ch )
m (cl ; ch )
Ei[cl ]) + e ch (Ei[ ch ] Ei[ cl ])
;
e (ch cl ) e (ch cl )
e ch (Ei[ch ] Ei[cl ]) + e ch (Ei [ cl ] Ei [ ch ])
;
e (ch cl ) e (ch cl )
e cl (Ei[ch ] Ei[cl ]) + e cl (Ei [ ch ] Ei [ cl ])
;
e (ch cl ) e (ch cl )
e cl (Ei[ch ] Ei[cl ]) + e cl (Ei [ cl ] Ei [ ch ])
; and
e (ch cl ) e (ch cl )
e (ch cl ) 1
2 (0; 1) :
1 + e (ch cl )
e
ch
(Ei[ch ]
Then the cash and capital values can be rewritten as
RC
RK
+
fl (cl ; ch ) ; q (ch ; cl ; ch ) =
2
2
cl RC
RC
RK
v (cl ; cl ; ch ) = RK +
+
m (cl ; ch ) +
2
2
2
ch RC
RC
RK
v (ch ; cl ; ch ) = RK +
m (cl ; ch ) +
2
2
2
q (cl ; cl ; ch )
=
41
RC
RK
+
fh (cl ; ch ) ;
2
2
gl (cl ; ch )
cl fl (cl ; ch ) ; and
gh (cl ; ch )
ch fh (cl ; ch ) :
For any ch , de…ne the function H (ch ) implicitly as the corresponding lower threshold cl so that at c = ch
the market price is just h, i.e.,
v (ch ; cl = H (ch ) ; ch )
= h:
q (ch ; cl = H (ch ) ; ch )
p (ch ; cl = H (ch ) ; ch ) =
Similarly, de…ne L (ch ) is de…ned implicitly by
v (cl ; cl = L (ch ) ; ch )
= l;
q (cl ; cl = L (ch ) ; ch )
p (cl ; cl = L (ch ) ; ch )
which makes the market price to be l at c = cl . Obviously, once we …nd such ch that H (ch ) = L (ch ), then
this particular ch and the corresponding cl = H (ch ) = L (ch ) is a solution of (7)-(9), (12)-(13). To show
that this solution exists, we …rst establish properties of L (ch ) then we proceed to the properties of H (ch ).
Properties of L (ch ) It is useful to observe that
@fl
@cl
=
@gl
@cl
=
lim fl
=
cl !ch
e2
(e2
ch
ch
+ e2
e2
cl
1
cl
fl
cl )
;
@fl
=2
@ch
e
e2 ch + e2 cl
1
@gl
+ 2 c
=
gl ;
cl
(e h e2 cl )
@ch
e
1
; lim gl = 0; lim m = 0:
cl !ch
ch cl !ch
1
ch
cl )
(ch
2 gh
e
(ch cl )
fh
e
(cl ch )
(cl ch )
;
1. We show that fl (ch ; cl ) is monotonically decreasing in cl : Its slope in cl is
e2
@fl
= 2
@cl
(e
ch
ch
+ e2
e2
cl
1
cl
fl (ch ; cl )
cl )
;
(B.12)
and the second derivative is
@ 2 fl
=
@ 2 cl
=
e2
2 ch
4 e
(e2
ch
cl
e2
cl )2
e2
ch
(e2
ch
+ e2
cl 2
e2
cl )2
!
1
cl
fl (ch ; cl )
=
1
cl
1
c2l
e2
(e2
fl (ch ; cl ) +
ch
ch
e2
(e2
+ e2
e2
ch
ch
cl
cl )
+ e2
e2
cl
cl )
=
1
c2l
Note that if the …rst derivative is zero, then the second derivative is positive implying that fl (ch ; cl )
can have only local minima, but no local maxima in cl . At the limit one can check that
@fl
lim
= lim
cl !ch @cl
cl !ch
1 e2
cl (e2
ch
ch
+ e2
e2
cl
cl )
( cl fl (ch ; cl )
!
1)
=
1
ch
1
2 ch
< 0:
Thus, fl (ch ; cl ) is decreasing at ch = cl : Suppose that it is not monotonic over the range of cl < ch in
cl : Then the largest c^l where the …rst derivative is 0, would be a local maximum. But we have just
ruled out the existence of a local maximum. Thus fl (ch ; cl ) monotonically decreasing over the whole
range of cl < ch in cl : This statement is equivalent to fl (ch ; cl ) c1l < 0 for cl < ch ; for any …xed ch :
42
2. We show that X (cl )
1
cl
fl (ch ; cl )
is increasing in cl . We would like to show that
e2
(e2
X 0 (cl ) =
ch
ch
+ e2
e2
cl
cl )
1
> 0:
c2l
X (cl ) +
(B.13)
Clearly, we have
1
1
=
> 0:
2
ch
2 c2h
X (cl = ch ) = 0, X 0 (cl = ch ) = fl0 (ch ; ch ) +
We know that when cl ! 0, f (ch ; cl ) has the order of Ei ( cl ) which is O (ln cl ); this implies that
X (cl ) ! 1 when cl ! 0. Then, if X (cl ) is not monotone, we must have two points x1 < x2 closest
to (but below) ch so that
0 > X (x1 ) > X (x2 ) , X 0 (x1 ) = X 0 (x2 ) = 0:
Setting (B.13) to be zero, we have (because 0 < x1 < x2 )
e2 ch
2 x2 (e2
1
X (x1 ) =
e2 x1
ch + e2
e2 ch
2 x2 (e2
2
<
x1 )
e2 x2
ch + e2
x2 )
= X (x2 ) ;
in contradiction with X (x1 ) > X (x2 ). Thus (B.13) holds always.
3. We show that the function gl (ch ;cl ) cl fl (ch ; cl ) is monotonically increasing in cl . Its …rst derivative
is (all the derivatives in this part are with respect to cl )
gl
cl fl
0
=
e2
1
+ 2
cl
(e
ch
=
e2
1
+
cl
(e2
+ e2
e2
ch
ch
ch
cl
g
cl ) l
+ e2
e2
cl
e2
(e2
(cl ; ch )
gl
cl fl
cl )
ch
ch
+ e2
e2
e2
(e2
+
ch
ch
!
cl
(cl fl (cl ; ch )
cl )
+ e2
e2
1) + fl (cl ; ch )
cl
fl
cl )
Whenever the …rst derivative is zero, at that point we have
gl
cl fl =
1
cl
fl
(e
(e2
2 ch
+e
ch
1
)
cl
)
2 cl
e2
:
(B.14)
We also know that
lim
gl
cl !ch
cl fl
0
gl
= 0; and lim
cl !ch
cl fl
00
so for any …xed ch ; cl = ch is a local maximum. Thus to show that
rule out the case of a local minimum c^l < ch so that
gl
cl fl
00
=
e2
1
+
c2l
(e2
Thus, if there were a c^l that
ch
ch
+ e2
e2
gl
cl
gl
cl )
cl fl
0
cl fl
gl
0
cl fl
fl0 +
0
1
< 0;
3 c2h
=
gl
cl fl is monotone, it su¢ ces to
= 0 and
4e2
(e2 ch
gl
ch 2 cl
e
cl fl
2
gl
2
e2 cl )
= 0, using (B.12) and (B.14) we have
43
00
gl
> 0. In general
cl fl
cl fl
+
00
1
:
to be
equal to
1
c^2l
fl0 +
2 2 ch 2 c^l
4
e
(e2
e
2
e2 c^l )
ch
0
1
c^l
fl
B
@
(e
(e2
2 ch
+e
ch
1
+
)
c
^l
)
2 c
^l
e2
1
1C
A=
e2
e2
1
c^2l
ch
ch
e2
+ e2
c^l
1
c^l
fl
c^l
:
But from (B.13) we know the above term is strictly negative, which proves the contradiction.
gl
4. We show that q (cl ; cl ; ch ) is also decreasing in cl for any cl < ch : Given that
cl RC
2
( ch
RC e
cl )
+e (ch
cl )
2
cl fl
0
> 0 and
2 ch +2 cl
e
+1
2 > 0; v (cl ; cl ; ch ) is increasing in cl :
(e ch + cl +1)
Thus, p (cl ; cl ; ch ) is increasing in cl for any cl < ch : Also one can show that limcl #0 = p (cl ; cl ; ch ) =
tanh( ch )
< 0; and
@
+
e (ch
2
cl )
e
(ch
=@cl = 12 RC
cl )
lim p (cl ; cl ; ch ) =
RK
2
RK + ch R2C +
RC
2
cl !ch
+
1
ch
ch
RK
=
1
ch
2
RK + ch R2C
RC
2
+
RK
2
RK
2ch
;
which is larger than l as long as ch > l: Thus, as long as ch > l; limcl !ch p (cl ; cl ; ch ) l and there is
a unique solution cl for any ch of p (cl ; cl ; ch ) = l. Therefore L (ch ) exist. From the monotonicity in
cl , and continuity of p (cl ; cl ; ch ) we also know that L (ch ) is continuous.
Properties of H (ch ) First, we show that for any ch 2 [l; RK ] ; H (ch ) is a continuous function and
H (ch ) 2 [0; ch ] : Again, the notation 0 means we are taking the derivative with respect to cl . We use the
following facts:
2
@fh
@cl
=
@fh
@ch
=
lim fh
=
cl !ch
1. The result of
2. We show
@2
gh
(ch cl )
e
;
e2 ch + e2 cl
1
(e2 ch e2 cl ) ch
1
; lim gh = 0:
ch cl !ch
@fh
@cl
gh
(ch cl )
e
1
cl
fl (ch ; cl )
=
fh ch
fh ch
=
@ 2 cl
2
e (ch
0
fl (ch ;cl )
cl )
e
1
cl
(ch
@gh
=
@cl
e
fh (ch ; cl ) ;
e
ch 2 fh0
(ch cl )
e
cl )
e2
(e2
@gh
1
=
@ch
ch
ch
+ e2
e2
ch
cl
gh (cl ; ch )
cl )
< 0 follows from the step 1 in the previous subsection.
cl )
> 0 for cl < ch . We have
2gl0
2 gl (ch ; cl )
e (ch
(ch cl )
2 cch2
l
(ch cl )
+ e
gh
fh ch
(ch cl )
e2(
e
0
=2
gl ch fl +ch c1
l
e (ch
(ch cl ))
cl )
+1
2 (ch cl )
1
(ch
e
2
cl )
2gl
and
ch 2 fl +
2ch
cl
If the …rst derivative is zero at a point ch > cl ; then the second derivative is
2 c1l + 2
(e2
(e2
ch
ch
+e2
cl
e2 c l
)
g (c ; c ) ch fl (ch ; cl ) +
) l l h
e (ch cl ) e (ch cl )
44
ch
cl
ch 2 c12
l
=
2 chc2 cl
l
e
(ch cl )
e
(ch cl )
< 0:
:
for any ch > cl ; which implies that it can have no minimum in that range. Also
gh
@
lim
@2
fh ch
= 0;
@cl
cl !ch
gh
lim
fh ch
so cl = ch must be the unique maximum in the range ch
1
3 c2h
=
@ 2 cl
cl !ch
cl , and the result follows.
3. Consequently, q (ch ; ch ; cl ) is monotonically decreasing and v (ch ; ch ; cl ) is monotonically increasing in
cl : Thus, p (ch ; ch ; cl ) is monotonically increasing in cl :
4. Observe that the following hold
lim p (ch ; cl ; ch ) = lim
cl !ch
cl !ch
RK
RK ch + c2h R2C
c2 RC + RK ch
v (ch ; cl ; ch )
2 ch
=
= h
= ch :
R
R
C
K
q (ch ; cl ; ch )
RC ch + RK
2 ch + 2
Because limcl !0 p (ch ; cl ; ch ) = ch , hence we know that for any ch > h there is a unique cl 2 [0; ch ]
which solves p (ch ; cl ; ch ) = h: From the monotonicity of p (ch ; ch ; cl ) in cl and the continuity in ch ; the
resulting function H (ch ) is continuous in ch .
Intercept of H (ch ) and L (ch )
1. Here we show that H (h) > L (h) : We know that H (h) = h because
lim =
cl !h
RK + h R2C + RK2
v (ch ; cl ; ch )
=
RK
RC
q (ch ; cl ; ch )
2
2 +
1
h
h
=
1
h
RK + h R2C +
RC
2
+
RK
2
RK
2
h
1
h
1
h
= h:
However, note that
RK + h R2C + RK
2
v (cl ; cl ; ch )
lim
=
R
R
C
K
cl !h q (cl ; cl ; ch )
2 + 2h
l ;cl ;ch )
and v(c
q(cl ;cl ;ch ) is increasing in cl : Since L (h) is de…ned by
hold.
h
v(cl ;L(h);h)
q(cl ;L(h);h)
1
h
= h;
= l < h; L (h) < h = H (h) must
2. Now we show that limch !1 H (ch ) = 0 < limch !1 L (ch ) : It is easy to check that
lim fl =
ch !1
Thus, limch !1
v(cl ;cl ;ch )
q(cl ;cl ;ch )
Ei [
Ei[ cl ]
; lim gl =
e ( cl ) ch !1
e (
= 0; lim gh = 0
ch !1
takes the value of
RK +
lim
cl RC
2
+
RC m(cl ;ch )
2
RC
2
ch !1
RK +
=
cl ]
; lim fh
cl )
ch !1
cl RC
2
+
RC
2
+
+
+
RK
2
Ei[
cl ]
e ( cl )
Ei[ cl ]
e ( cl )
RK
2
RC
2
gl (cl ;ch )
RK
2
cl fl (cl ; ch )
fl (cl ; ch )
cl
Ei[ cl ]
e ( cl )
:
Thus, limch !1 L (ch ) is the …nite positive solution of
RK +
cl RC
2
+
RC
2
+
RK
2
RC
2
Ei[
e (
Ei[ cl ]
e ( cl )
45
cl ]
cl )
cl
Ei[ cl ]
e ( cl )
= l:
In contrast, limch !1
RK +
lim
v(ch ;cl ;ch )
q(ch ;cl ;ch )
ch RC
2
takes the value of
RC
2 m (cl ; ch )
RC
2
ch !1
=
Hence,
lim
RK
ch
ch !1
v(ch ;cl ;ch )
q(ch ;cl ;ch )
+
RC
2
RC
ch 2
+
RC
2ch
+
RK
2
ch fh (cl ; ch )
fh (cl ; ch )
gh (cl ;ch )
ch
RK
2
+
gh (cl ;ch )
RK
2
+
fh (cl ; ch )
= lim
fh (cl ;ch )
ch
RK
2
ch !1
v(ch ;cl ;ch )
q(ch ;cl ;ch )
h ;cl ;ch )
limch !1 v(c
q(ch ;cl ;ch ) = l;
grows without bound for any …xed cl , and
cl . As a result, in order to have a solution of
limch !1 H (ch ) = 0:
RC
2
+
RK
2
RK
2
gh (cl ;ch )
ch
fh (cl ;ch )
ch
= 1;
is monotonically increasing in
cl has to go to zero, implying
The two results imply that there is always an intercept ch 2 (h; 1) that H (ch ) = L (ch ) : This concludes
the step proving that (7)-(9), (12)-(13) has a solution.
B.1.3
Step 3: Proof of Lemma B.4
We have shown that H (h) = h: Note also that if ch = cl then vqhh = vqll : This, and the continuity of H ( ) and
L ( ) in l; implies that at the limit l ! h; there is a solution of the system (7)-(9), (12)-(13) that cl ch ! 0
and ch ; cl ! h: Then, the statement comes from h < hRC < RK (as RC > 1).
B.1.4
Step 4: Proof of Lemma B.5
First we show that q (c) is always deceasing, and there exists a critical value b
c 2 (cl ; ch ) so that q 00 (c) < 0 for
00
00
c 2 (cl ; b
c) and q (c) > 0 for c 2 (b
c; ch ). Moreover, for c 2 (cl ; b
c) where q (c) < 0, we have that q 000 (c) > 0.
1. To show that q 0 < 0, we di¤erentiate the ODE 0 =
2
0=
2
q 000
2
2
q 00 +
RK
2 c2
2
RC +
q0 :
RK
c
q again to reach
(B.15)
Due to boundary conditions, we have at both ends cl and ch , the function q 0 (c) equals zero and
2
its second derivative 2 q 000 = 2 RcK
> 0. Suppose to the contrary that q 0 (e
c) > 0 for some point
2
e
c 2 (cl ; ch ); then we can pick e
c so that q 0 (e
c) > 0 and q 000 (e
c) = 0 (otherwise the function q 0 ( ) is zero
at one end, is convex globally, and thus never comes back to zero at the other end). But because
2
000
0
c) = 2 RbcK
c) > 0; contradiction. This proves that q 0 < 0.
2 + q (e
2 q (e
2. We know that q 00 (cl ) < 0 and q 00 (ch ) > 0, and therefore there exists b
c so that q 00 (b
c) = 0: We show
2
2
RK
00
q, we have 0 = 2 q 000 2 RcK
q 0 , and
this point is unique. Because 0 = 2 q + 2 RC + c
2
2
0=
2
q 0000 +
RK
c3
q 00 .
(B.16)
Suppose we have multiple solutions for q 00 (b
c) = 0. Clearly, it is impossible to have q 00 (b
c) = 0 but
00
00
0000
q (b
c ) > 0 and q (b
c+) > 0; otherwise q (b
c) > 0 which contradicts with (B.16). Then there must
exist two points c1 > b
c and c2 > c1 > b
c that q 00 (c1 ) = 0, q 00 (c2 ) < 0 and q 0000 (c2 ) > 0; but q 00 (c) < 0
2
RK
0000
for c 2 (c1 ; c2 ). This implies that 2 q (c1 ) =
+ q 00 (c1 ) < 0: As a result, there exists another
c31
point c3 2 (c1 ; c2 ) so that q 0000 (c3 ) = 0 with q 00 (c3 ) < 0. But this contradicts with (B.16).
3. Now we show that for c 2 (cl ; b
c) with q 00 (c) < 0, we have q 000 (c) > 0, i.e., q 00 (c) is increasing. Suppose
000
not. Since q (cl ) > 0 so that q 00 (c) is increasing at the beginning, there must exist some re‡ecting
point c4 for the function q 00 so that q 0000 (c4 ) = 0. But because q 00 (c4 ) < 0, it contradicts with (B.16).
46
Second, we show that v (c) is increasing if h
l is su¢ ciently small.
1. We show that if v 00 (cl ) > 0; then v (c) is increasing in c. Let F (c)
0 = q 00
2
2
+
2
F 00 + RC
2
v 0 (c), so that
F
with boundary conditions that F (cl ) = F (ch ) = 0. The assumption v 00 (cl ) > 0 implies that F 0 (cl ) >
0. Thus, if there are some points with F (c) < 0 in the range of (cl ; ch ) then we can …nd two points c1
and c2 (a maximum and a minimum) so that c1 < c2 but F 00 (c1 ) < 0 F 00 (c2 ) > 0, F 0 (c1 ) = F 0 (c2 ) = 0
and F (c1 ) > 0 > F (c2 ). We can apply the ODE to these two points:
0
= q 00 (c1 )
2
0
= q 00 (c2 )
2
2
+
2
2
+
2
F 00 (c1 ) + RC
2
F (c1 ) ;
F 00 (c2 ) + RC
2
F (c2 ) :
The second equation implies that q 00 (c2 ) < 0, which implies that c1 < c2 < b
c. However, the above two
equations also imply that
q 00 (c1 )
2
>
2
RC > q 00 (c2 )
2
contradiction with the previous lemma which shows that q 00 is increasing over [cl ; b
c] :
l is su¢ ciently small, then v 00 (cl ) > 0; with the …rst result we obtain our
2. Now we show that if h
claim. From our ODE,
v 00 (cl ) =
2
2
(RC cl + RK )
2
We know that as h
limcl !ch (RC cl2+RK )
lim
cl !ch
RK
RC
RK
+
h (cl ; ch ) +
2
2
2
(RC cl +RK )
@
2
RC e
lim
RC
1
2
(1 + 1)
2
RK
RC
RK
+
h (cl ; ch ) +
2
2
2
+
+1
RK
2
gl (cl ; ch )
cl fl (cl ; ch )
@
= limcl !ch
(ch cl )
(ch cl )
e
2
v(cl )
@cl
cl !ch
2
1
ch
RC
2
e2
1
+ 2
cl
(e
RK
+
2
These two statements imply that if ch
B.2
=
gl (cl ; ch )
cl fl (cl ; ch )
l ! 0; ch
cl ! 0: We will prove the statement by showing that (1)
v (cl ) = 0, because limcl !ch (RC cl2+RK ) v (cl ) equals
and (2) limcl !ch
it equals
=
v (cl )
1
2 ch
1
2 ch
h(cl ;ch )+
=
RK
2
RK
RK
+0+
2
2
gl (cl ;ch )
ch
+ e2
e2
=
cl
g
cl ) l
=0
cl fl (cl ;ch )
< 0, because
@cl
ch
1
0
e2
(e2
ch
ch
+ e2
e2
cl
cl )
(cl fl
!!
1)
RC
< 0:
4
cl is small enough then v 00 (cl ) > lim v 00 (cl ) = 0.
cl !ch
Proof of the Second Part of Proposition 5
(cl ;ch )
The result ch > h is a consequence of the fact that we de…ned H (ch ) as the unique cl solving vqhh (c
=h
l ;ch )
when ch > h: (see part 4 in section B.1.2.)
For the result cl
l; consider the possibility that cl > l: The following lemma states that in this case
p00 (cl ) < 0: This implies that this is not an equilibrium. To see this, we have p0 (cl ) = 0 by the boundary
47
:
conditions v 0 (cl ) = q 0 (cl ) = 0. Thus p00 (cl ) < 0, combined with p (cl ) = l and p0 (cl ) = 0, would imply that
p (c) < l for c su¢ ciently close to cl :
Lemma B.6 The sign of p00 (cl ) is the same as that of l
cl :
Proof. Simple algebra implies that
p00 (cl )
v0 q
=
=
0
q2
v 00 q
q 00 v
q2
2
=
=
(v 00 q + v 0 q 0
3
2q
q2
2
=
(q 00 v + v 0 q 0 ))
2
(RC cl + RK ) + lq (cl )
2
(v 0 q
q
q 0 v)
2
(RC cl + RK ) + cl q (cl )
2
q
2
v
2c
l
q2
(RC cl + RK ) + lq (cl ) + cl q (cl )
cl q (cl )
2
2
(RC cl + RK ) + cl q (cl )
q2
2
=
=
q0 v
2
(RC cl + RK ) + cl q (cl )
v
cl
q
2
+ (l
cl ) q (cl )
2
2
q
q2
(l
cl )
1
cl
2
(RC cl + RK )
2
cl q (cl )
2
+ q (cl )
2
2
q
which gives the lemma by noticing that q is decreasing in c and the boundary q 0 (cl ) = 0 implies that
2
(RC cl + RK ) + cl q (cl ) / q 00 (cl ) < 0:
The third statement is a consequence of the following Lemma.
Lemma B.7 We have the following limiting results:
lim
!1
1
1
; lim fh = , lim gh = 0, lim gl = 0;
!1
!1
cl
ch !1
= h, lim cl = l:
fl
=
and lim ch
!1
!1
Proof. The …rst four results are based on L’Hopital rule. Take the …rst result for illustration:
lim
!1
fl
=
lim
!1
(Ei[ ch ]
e (
e
=
lim
!1
1
2
e
Ei[ cl ])
cl )
ch
e
( cl )
+
= lim
!1
Ei[ ch ]
1
e
cl
( cl )
e
( cl )
= lim
e
cl
Ei[ cl ]
( cl )
=
!1 ( cl ) e ( cl )
=
1
:
cl
These four results imply that
RK +
vh
lim
= lim
!1 qh
!1
ch RC
2
Thus, in the limit the solution of
RC
2 m (cl ; ch )
RC
2
vh
qh
+ RK 2
gh (cl ;ch )
ch fh (cl ; ch )
=
+ RK 2 fh (cl ; ch )
= h is the solution for the equation of
RK +
RC
2
ch RC
2
RK 12
+ RK 2c1h
48
= h;
RK +
RC
2
ch RC
2
RK 12
+ RK 2c1h
2
2c
l
v
which gives lim
!1 ch
= h: Similarly, the following calculation implies that lim
RK +
vl
lim
= lim
!1 ql
!1
B.3
cl RC
2
+
RC
2 m (cl ; ch )
RC
2
+ RK 2
gl (cl ;ch )
!1 cl
cl fl (cl ; ch )
=
+ RK 2 fl (cl ; ch )
= l:
RK +
RC
2
cl RC
2
+ RK 12
+ RK 2c1 l
:
Proof of Proposition 7
The proofs of the two statements follow the same logic. Thus, we prove the …rst statement in detail and
explain the necessary modi…cations for the second statement at the end of the proof.
Consider the functions q~ (c; q0 ; v0 ; ch ) and v~ (c; q0 ; v0 ; ch ) of c parameterized by q0 ; v0 ; and ch :
2
q~00 (c) +
0
=
0
= q~0 (c)
2
2
2
2
+
2
(RC
q~ (c)) +
v~00 (c) +
2
2
(RC c
RK
c
q~ (c)
v~ (c)) +
2
(RK
(B.17)
v~ (c)) :
(B.18)
and the boundary conditions
v~0 (ch )
q~ (c0 )
= q~0 (ch ) = 0;
= q0 ; v~ (c0 ) = v0 :
(B.19)
(B.20)
The general solution is
RC
RK
ec Ei ( c) + e c Ei (c )
+ e c A1 + ec A2 +
2
2
2
cRC
cRK e c Ei (
ve (c) = RK +
+ ec (A3 cA2 ) e c (A4 + cA1 ) +
2
2
= RK + RC c + ec A3 e c A4 ce
q (c) :
qe (c)
=
(B.21)
c)
e
c
Ei ( c)
2
(B.22)
(B.23)
where A1 -A4 (may di¤er from those in (12) and (13)) are pinned down by (B.19)-(B.20). We have
qe0 (c)
ve0 (c)
Ei[c ] + ec Ei[ c ])
;
2
c
c
2
c
RC
RK ( e
Ei[c ] + e Ei[ c ]) RK c (e
Ei[c ] + ec Ei[ c ])
=
+
+
2
2
2
+ec (( c 1) A2 + A3 ) + e c (( c 1) A1 + A4 ) :
=
e
c
A1 + ec A2
RK
2
(e
c
De…ne the function ch (q0 ; v0 ) implicitly by v~ (ch ; q0 ; v0 ; ch ) = h~
q (ch ; q0 ; v0 ; ch ), and we are interested in
the derivatives
v~q0 0 h~
qq0 0 @ch
v~v0 0 h~
qv0 0
@ch
=
;
=
:
@q0
v~c0 h h~
qc0 h @v0
v~c0 h h~
qc0 h
@ch
h
We proceed as follows. First we show that @c
q (c0 ) and
@q0 > 0 and @v0 < 0: This proves that if q (c0 )
v (c0 ) v (c0 ) then ch < ch ; that is, such policies make the overinvestment problem worse. Then we show
that this is true even if q (c0 ) q (c0 ) and v (c0 ) v (c0 ) ; as long as the policy increases the price at c0 ;
(c0 )
0)
i.e. vq (c
> v(c
q(c0 ) .
0)
We start with the following lemmas.
49
Lemma B.8 We have
@ q~ (ch ; q0 ; v0 ; ch )
@q0
@~
v (ch ; q0 ; v0 ; ch )
@v0
e
=
2
+e
2
c0 ) + e
ech e
=
c0
(ch
e
ch
> 0;
c0
e
(ch c0 )
> 0;
(B.24)
@ q~ (ch ; q0 ; v0 ; ch )
= 0:
@v0
(B.25)
Proof. We show (B.24) …rst. We know that q~ (c0 ) = q0 , which based on (B.21) can be written as
A1 + e c0 A2 + lq = q0 (where lq is independent of q0 ) which implies
c0
lq
A1 =
and q~0 (ch ) = 0 which can be rewritten as
which implies
A2 =
ch
e
A1
sq
ech
=
e
ch
e
e
e
A2 + q0
c0
A1 + ech
lq e c0 A2 +q0
e c0
ech
ch
c0
sq
:
(B.26)
A2 + sq = 0 (where sq is independent of q0 )
) A2 =
e ch
(1 + e
lq +q0
sq
e c0
2ch e 2c0 ) ech
:
(B.27)
Thus, (B.27) and (B.26) imply that
@A2
@q0
@A1
@q0
=
=
e
ech e
1
e c0
c0
ch
+e
e
ch
;
c0
e
(B.28)
ch
e
2c0
ech e
c0
ch
+e
e
c0
=
ech
ech
e c0 + e
ch
e
c0
:
(B.29)
Using (B.21) we obtain our result.
0 ;v0 ;ch )
The …rst result in (B.25) follows similarly. The second result @ q~(ch ;q
= 0 comes from the fact that
@v0
(B.17) and the boundary conditions q~0 (ch ) = 0 and q~ (c0 ) = q0 are independent of v0 .
Lemma B.9 We have
@~
v (ch ; q0 ; v0 ; ch )
@q0
@~
v (ch ; q0 ; v0 ; ch )
@q0
h
@ qe (ch ; q0 ; v0 ; ch )
@q0
=
2
=
2
e
(ch c0 )
e
(ch c0 )
(e
e
(ch c0 )
e
(ch
c0 e
ch
(ch c0 )
(e
ch
+e
(ch c0 )
< 0;
c0 e ch )2
+e
(ch + h
c0 e
(ch c0 )
c0 ) e
+e
c0 ) e
(ch c0 )
+e
(ch c0 )
c0 e ch )2
Proof. We show the …rst result. We rewrite ve (c0 ) and ve0 (ch ) as (as before here lvq and svq are
independent of q0 )
ve (c0 ) = ec0 (A3 c0 A2 ) e c0 (A4 + c0 A1 ) + lvq ;
ve0 (ch ) = svq + ech (( ch 1) A2 + A3 ) + e ch (( ch
1) A1 + A4 )
Thus, the boundary conditions ve (c0 ) = v0 and ve0 (ch ) = 0 imply that
A3
= c0 A2 + e
( e
ch
c0
v0
( ch
e
c0
=
2c0
c0 + 1)) A2 + e
+( e
A4
lvq + e
c0
e
ch
e
(A4 + c0 A1 ) ;
ch
( ch
) v0 + (sqv
ch
50
+ e
1) + c0 e
e
2 c0 e ch
c0
e
ch
lqv )
2 c0
e
ch
A1
<0
Thus, using the result in (B.28) and (B.29) one can derive that
@A4
=e
@q0
ch
c0
2e
e
ch
c0 (e
ch
c0 e
(e
c0
e
+e
ch
c0
e
ch
+ c0 e
c0
+e
)
2
c0 e ch )
:
Similarly it implies that
@A3
@A1
=
e
q0
q0
2c0
c0 +
@A2
@A4
c0 +
e
q0
q0
2c0
=
2e
c0
(e
ch
+e
(ch c0 )
+e
c0 e
2 ch
e
c0
+e
c0 e ch )2
Consequently, using (B.23), we have (where we have used (B.24))
@e
v (ch )
@q0
= ech
=
2
e
@A3
q0
(ch c0 )
@A4
q0
ch
e
ch
(ch c0 )
e
(e
@ qe (ch )
@q0
(ch c0 ) e
c0 e
ch
+e
(ch c0 )
< 0:
2
c0 e ch )
The last inequality comes from the fact that the function ex e x x (e x + ex ) is negative and monotonically
(ch )
:
decreasing for all x > 0: The second statement comes directly from the expression for @ qe@q
0
Lemma B.10 If
v0
q0
< h; then v~ (y; q0 ; v0 ; y)
h~
q (y; q0 ; v0 ; y) > 0:
Proof. We parameterize ch by y. The idea is that if the function v~ (y; q0 ; v0 ; y) h~
q (y; q0 ; v0 ; y) is
negative at y = c0 and positive as y ! 1; then there is a y = ch so that this function is zero (satisfying the
de…nition of ch ) and where the slope of this function is positive, which is the claim of our lemma.
The function v~ (y; q0 ; v0 ; y) h~
q (y; q0 ; v0 ; y) can be solved by imposing the boundary conditions
v~0 (y) = q~0 (y) = 0; q~ (c0 ) = q0 ; v~ (c0 ) = v0 :
for all y
(B.30)
c0 . Thus, by setting y = c0 , we must have
v~ (c0 ; q0 ; v0 ; c0 )
h~
q (c0 ; q0 ; v0 ; c0 ) = v0
hq0 < 0;
by the condition of the proposition.
Now we show that v~ (y; q0 ; v0 ; y) h~
q (y; q0 ; v0 ; y) ! 1 as y ! 1. We …rst show calculate limy!1 q~ (y; q0 ; v0 ; y)
in (B.21). For this, we solve for e y A1 and ey A2 from (B.21)-(B.22) and (B.30):
e
y
A1 =
q0
RC
2
+ e(c0
e(y
0
RK
y) RK M (y)
2
2
c0 ) + e (c0 y)
M (c0 )
y
; e A2 =
q0
RC
2
0
(y)
e(y c0 ) RK M
2
e(y c0 ) + e (c0 y)
RK
2
M (c0 )
:
where M (y)
e y Ei [ y]+e y Ei [y ]. Using limy!1 M 0 (y) = 0, it is easy to show that limy!1 ey A2 =
y
limy!1 e
A1 = 0, which implies that limy!1 q~ (y; q0 ; v0 ; y) = R2C in (B.21). A similar argument implies
that limc!1 v~ (c; q0 ; v0 ; c) = 1. Thus, v~ (c; q0 ; v0 ; c) h~
q (c; q0 ; v0 ; c) = 1. This prove the statement.
Putting together the above three lemmas, we have
v~q0 0
v~c0 h
@ch
=
@q0
h~
qq0 0
@ch
> 0; and
=
h~
qc0 h
@v0
This implies that ch < ch whenever q (c0 )
For the last step, as
@ch
@v0
=
v
~v0 0
v
~c0
h
h~
qv0 0
h~
qc0
q (c0 ) and v (c0 )
v~v0 0
v~c0 h
h~
qv0 0
<0
h~
qc0 h
v (c0 ).
< 0, it su¢ ces to show that this result holds for the worst v0 drop
h
to maintain p0 , i.e., v0 and q0 decrease proportionally so v0 =q0 remains at constant.
51
To this end, we consider decreasing q0 to q0 = q0 " where " is very small. To make sure that vq00 = vq00 ;
we need that v0 = v0 a" where a = vq00 . Let us refer to all the objects after the change with the bar. Our
goal is to show that v (ch ) =q (ch ) would increase; then v~c0 h h~
qc0 h > 0 implies that ch < ch . Using the …rst
two Lemmas above, we have (denoting x (ch c0 ) )
q (ch )
= q~ (ch )
"
2
+e
ex
x
v (ch )
= v~ (ch )
2"
e
x
x
e
x
x (e
(ex
+e
+ ex )
v0
2"
q0 ex + e
x )2
x
Hence for su¢ ciently small " we have (up to the …rst order)
v (ch )
q (ch )
ex
=
v~ (ch )
q~ (ch )
2"
q~ (ch )
=
v~ (ch )
q~ (ch )
0
2" @ ex
q~ (ch )
e
x
(ex + e
e
x
(ex
x
x (e
x )2
x
x (e
+e
+ ex )
+ ex )
x )2
Here, the third inequality in (B.31) is because the term ex e
x
v0
1
+
q0 ex + e
+
v
~(ch )
q~(ch )
v0
q0
ex
x (e
x
x
+e
x
!
+
v~ (ch )
2"
q~2 (ch ) ex + e
x
1
A > v~ (ch ) :
q~ (ch )
+ ex ) < 0 for all x > 0 and
(B.31)
v0
q0
v
~(ch )
q~(ch )
(ch )
is strictly negative because vq00 < vq~~(c
= h; hence the …rst order impact of decreasing q0 is an increase in
h)
v (ch ) =q (ch ). Because the above argument holds for any v0 and q0 , tracing out the …rst-order e¤ect implies
h)
that any intervention which lowers cash value but keeps capital price unchanged will lower v(c
q(ch ) . Compared
to that change, an increase in v0 just decreases ch further. That concludes our proof.
The second statement follows the same steps with the following modi…cations. Each ch has to be changed
to cl and each h has to be changed to l at every point of the proof: Then the …rst lemma remains the same,
@v
~(cl ;q0 ;v0 ;cl )
the …rst statement in the second lemma changes to
> 0, while the second statement does not
@q0
change. Also, in the proof of the …rst statement we use that ex e x x (e x + ex ) > 0 for all x < 0, and
the proof of the second statement we use that ex e x (x + y) (e x + ex ) < 0 for all x < 0 and y > 0: In
the last part we follow the same steps, but the inequality (B.31) in the modi…ed version is switched. This
gives that cl > cl under the conditions of the statement.
B.4
B.4.1
Solution for Price Floor Policy and Proof of Proposition 8
Characterizing the equilibrium with price ‡oor policy
We …rst derive the solutions for price ‡oor policy. A price ‡oor policy
0
= q 0 (c)
2
0
=
2
q 00
2
2
+
2
v 00 (c)
q (c) +
2
v (c) +
RC +
RK
c
2
(c) is de…ned as
(RC c + RK ) + c (c) ;
(c) :
(B.32)
(B.33)
so that 1) for c 2 (c0 ; cgh ], (c) = 0, and at the upper investment threshold p (cgh ) = h; and 2) for c 2 [cgl ; c0 ],
v (c) = (l + ) q (c) always. Here, v (c), q (c), (c), c0 and cgh are endogenous. We have the following lemma.
Lemma B.11 Given the lower disinvestment threshold cgl , the solution to the price ‡oor policy can be
calculated as follows.
1. Given the upper investment threshold cgh , …rst calculate the welfare function jg (c) = RK + RC c +
52
D1 e
c
+ D2 e c , where the constants D1 -D2 are given by the boundary conditions
jg (cgl ; cgh ) = (cgl + l) jg0 (cgl ) ; and j (cgh ; cgh ) = (cgh + h) jg0 (cgh ) :
2. For c 2 (c0 ; cgh ], the capital price and cash price is given by
v (c)
= RK +
q (c)
=
RC c
+ ec (A3
2
RC
+e
2
c
cA2 )
A1 + ec A2 + RK
e
c
(A4 + cA1 ) + cRK
ec Ei (
c) + e
2
2
c
c
(e
2
Ei (c )
Ei (
c)
e
c
Ei ( c))
2
;
:
Here, A4 = D1 and A3 = D2 . The other four constants, i.e., A1 -A2 , c0 and cgh , are determined by
the following four boundary conditions
v 0 (cgh ) = 0; q 0 (cgh ) = 0; v (c0 ) = (l + ) q (c0 ) ; v 0 (c0 ) = (l + ) q 0 (c0 )
3. For c 2 [cgl ; c0 ], we have
q (c) =
l+
jg (c)
and v (c) =
jg (c)
l+c
l+c
(B.34)
and the taxation is given by
2
(c) =
2
q 00
q (c) +
2
RC +
RK
c
>0
Proof. The total welfare function j (c) = v (c) + cq (c) given in the step 1 of Lemma B.11 only depends
on the investment/disinvestment policies cgl and cgh (see explanations around equation (18) and (19)). For
c 2 (c0 ; cgh ], there is not taxation and the derivation is the same as before, except that at the endogenous
intervention point c0 we are value-matching and smooth-pasting so that the price is the implemented ‡oor
price l + . Note that by construction we have v (cgh ) = hq (cgh ) (due to j (cgh ) = (cgh + h) j 0 (cgh )). For c 2
[cgl ; c0 ], notice that v (c) = (l + ) q (c) always; (B.34) follows because of jg (c) = v (c) + cq (c) = (l + c) q (c).
The endogenous taxation (c) follows from (B.33).
B.4.2
Proof of Proposition 8
Now we set
= 0 and prove Proposition 8. There are three steps.
Step 1. Rewrite the problem Clearly, for c 2 (c0 ; cgh ] the same structure solution applies without
policy, with the only di¤erence at the lower end c0 so that v 0 (c0 ) = lq 0 (c0 ) might not be zero. This allows us
to draw connection between the equilibrium with policy and the one without. We …rst show that for cgl < cl ,
the resulting slope at c0 has to be negative, i.e.
v 0 (c0 ) = lq 0 (c0 ) < 0:
To show this, focus on c 2 [cgl ; c0 ]. By v (c) =
v 0 (cgl ) =
l
l+c jg
(B.35)
(c) and boundary condition of jg (c), we have
l jg0 (cgl ) (l + cgl )
jg (cgl )
2
(l + cgl )
53
= 0:
0
Moreover, since jg00 (c) < 0 (see Proposition 2 and its proof), we have jg0 (c) (l + c)
0. As a result, since c0 > cgl , we have
sign [v 0 (c0 )] = sign jg0 (c0 ) (l + c0 )
jg (c) = jg00 (c) (l + c) <
jg (c0 ) < 0:
This proves (B.35).
This suggest us to introduce fv ( ) ; q ( ) ; c0 ; cgh ; xg indexed by x as the solution to the ODE system (10)
and (11), with modi…ed boundary conditions
v 0 (cgh ) = q 0 (cgh ) = 0; v (cgh ) = hq (cgh ) ;
v 0 (c0 ) =
xl; q 0 (c0 ) = x; v (c0 ) = lq (c0 ) :
Here, the parameter x > 0 captures the negative slope of v 0 (c0 ) = lq 0 (c0 ) < 0. As shown shortly, our
key result does not depend on the exact value of x, which will be determined by pre-determined lower
disinvestment threshold cgl .
It is easy to show that if cgl = cl , i.e., the policy sets the lower disinvestment threshold as the one in
the market solution, then x = 0 and we have c0 = cl = cgl and cgh = ch . Given this result, the claim in
Proposition 8 is equivalent to show that
@ch
> 0:
lim
!1 @x
Step 2. Solve the new ODE system For simplicity, we denote cgh by ch . Given c0 and ch , the
boundary conditions v 0 (ch ) = q 0 (ch ) = 0 and v 0 (c0 ) =
xl; q 0 (c0 ) =
x imply that
x e2
(e2
ch
q (c0 ; c0 ; x; ch )
= q (cl ; cl ; ch ) jcl =c0 +
q (ch ; c0 ; x; ch )
= q (ch ; cl ; ch ) jcl =c0 +
v (c0 ; c0 ; x; ch )
= v (cl ; cl ; ch ) jcl =c0
v (ch ; c0 ; x; ch )
= v (ch ; cl ; ch ) jcl =c0 +
c0
+ e2
ch
e2
c0 )
2xe (c0 +ch )
;
(e2 ch e2 c0 )
x e2 c0 ( l + 1) + e2 ch ( l
+
2 (e2 ch
e2 c0 )
2xe
(c0 +ch )
(c0
(e2 ch
ch +
2
e c0 )
l)
1)
;
:
where q (cl ; cl ; ch ) ; q (ch ; cl ; ch ) ; v (cl ; cl ; ch ) ; v (ch ; cl ; ch ) has been de…ned above. Then, c0 and ch solve
Fh (c0 ; x; ch ) = Fl (c0 ; x; ch ) = 0 where we de…ne
Fh (c0 ; x; ch ) v (ch ; c0 ; x; ch ) hq (ch ; c0 ; x; ch )
RC
RK
gh (c0 ; ch )
(ch h) RC
m (c0 ; ch ) +
= RK +
2
2
2
2xe (c0 +ch )
; and
(e2 ch e2 c0 )
Fl (c0 ; x; ch ) v (c0 ; c0 ; x; ch ) lq (c0 ; c0 ; x; ch )
(c0 l) RC
RC
RK
gl (c0 ; ch )
= RK +
+
m (c0 ; ch ) +
2
2
2
+
+
2xe
(c0 +ch )
(e2 ch
x e2
c0
(c0
ch +
e2 c0 )
l)
(ch + h) fh (c0 ; ch )
h
( l + 1) + e2 ch ( l
2 (e2 ch
e2 c0 )
1)
54
l
x e2
(e2
c0
+ e2
ch
e2
ch
(c0 + l) fl (c0 ; ch )
!
c0 )
Simple derivation reveals
@Fl
@c0
@Fh
@c0
=
=
2xe (c0 +ch )
+
(e2 ch e2 c0 )
@Fl
@ch
@Fh
@ch
=
=
e2
1
+ 2
c0
(e
RC 2e(c0 +ch )
RK
+
2 (ec0 + ech )2
2
0
RK @
RC 2e(c0 +ch )
+
2 (ec0 + ech )2
2
e
RC
2
ch
+ e2
e2
e
(ch
2gh
c0 )
e
g
c0 ) l
fl
2
(ch + h)
RC 2e
2 (ec0 + ech
2xe (c0 +ch )
(e2 ch e2 c0 )
2 (c0 + l)
1
ch
e
(ch
@ch
=
lim
!1 @x
e
fl
c0 )
(ch c0 )
1
A
1
ch
c0 )
fh
e
(c0 ch )
!
c0
)
g (c ; c )
c0
) h 0 h
fh (ch ; c0 )
Step 3. Prove the claim Now we are ready to show our desired result lim
is easy to show that when
theorem) implies
ch
c0
1
c0
fl
(ch c0 )
+ e2
e2
+1
(e2 ch +e2
RK B
(e2 ch e2
@
2 ch
2 c0
2 +
e
+e
(
)
2
)
(ch + h) e2 ch e2 c0 c1h
(
)
!
2 c0
2 ch
(c0 ch + l h) e
+e
+1
(e2 ch e2 c0 )
(c0 +ch )
e
ch
!
ch
(c0 ch )
0
e2
(c0 + l) 2
(e
c0
2gl (ch ; c0 )
(ch c0 )
e (ch c0 )
ch + l h) e2 c0 + e2
(e2 ch e2 c0 )
(c0
RK
RC 2e(c0 +ch )
+
2
c
c
2 (e 0 + e h )
2
RC
2
ch
fh (c0 ; ch )
1
C
A
@ch
!1 @x
> 0. First of all, it
! 1, ch ! h and c0 ! l are bounded. The Cramer’s rule (or implicit function
lim
!1
@Fh
@x
@Fh
@c0
@Fl
@x
@Fl
@c0
,
@Fh
@ch
@Fh
@c0
@Fl
@ch
@Fl
@c0
= lim
!1
@Fh @Fl
@x @c0
@Fh @Fl
@ch @c0
@Fh @Fl
@c0 @x
@Fl @Fh
@ch @c0
+
:
Focus on the denominator …rst. It is easy to show that
@Fl
RK l
RC
+
;
=
!1 @c0
2
2 c20
@Fh
RK h
@Fh
RC
@Fl
+
; and lim
=
= lim
= 0;
2
!1 @ch
!1 @c0
!1 @ch
2
2ch
lim
lim
implying
@ch
lim
=
!1 @x
For the numerator, since
@Fl
@x
=
1
2
and
@Fh
@x
@Fh @Fl
@c0 @x
lim
!1
RC
2
+
=
RK h
2c2h
@Fh @Fl
@x @c0
RC
2
+
(B.36)
RK l
2 c20
2e(c0 +ch ) (h l+(ch c0 ))
;
(e 2ch e 2c0 )
we can show the following two
limiting results:
lim
!1
e
(ch c0 )
e
(ch c0 )
@Fh @Fl
=
@x @c0
2 (h
55
l + (ch
c0 ))
RC
RK l
+
2
2 c20
;
(B.37)
1
c0
!
and
lim
!1
=
lim
!1
e
0
1B
@
(ch c0 )
e
(ch c0 )
c
c
RC 2(e h e 0 )
2 (ec0 +ech )
+
+
2x
RK
2
@Fh @Fl
@c0 @x
2gl (ch ; c0 )
(c0 ch +l h)(e2
(
e2 c h
c0
e2 c 0
(ch + h) 2
+e2
ch
)
)
1
c0
fl
+1
1
C
A = 0:
Hence, applying (B.37) and (B.38) to (B.36), we have
lim
!1
e
(ch c0 )
e
(ch c0 )
2 (h
@ch
=
@x
QED.
56
l + (ch
RC
2
+
RK h
2c2h
c0 ))
RC
2
RC
2
+
+
RK l
2 c20
RK l
2 c20
> 0:
(B.38)
C
Additional Material for He and Kondor (2015)
C.1
What if cPh > RK ?
Throughout the main body of the paper we have restricted our attention to the case where the equilibrium
range of cash-to-capital ratio, which is [cl ; ch ], is below RK . This ensures that in the idiosyncratic stage the
price pb = c
RK clears the market in a way that cash (capital) …rms get all the cash (capital). Otherwise,
suppose that ch > RK . Then, along the equilibrium path it is possible that c > RK , and the price of capital
at the idiosyncratic stage will be capped at the capital’s …nal output RK . As a result, cash …rms sell all their
capital K2 to capital …rms at a price of RK , ending up with a total amount of cash of C +R2 K K ; while the
capital …rms will have K units of capital but with C R2 K K units of cash in their hands. This allocation
is ine¢ cient as capital …rms are holding cash.
This concern is also relevant in the planner’s constrained e¢ cient allocation. Recall that overinvestment
requires the planner’s upper investment threshold cP
h > ch , hence it is quite likely that in a wide range of
parameters cP
h > RK . Because the planner is facing the same information constraint, i.e., the planner cannot
tell a cash …rm from a capital …rm, and the constrained outcome at the idiosyncratic stage is likely to be
ine¢ cient.
To illustrate that our main results hold in this case, in this Appendix we relax our parameter restriction
to
RK > lRC , and RK < h:
(C.39)
Note that we are replacing RK < hRC by RK < h; it just says that on the margin capital is better than
cash if we just consume cash (for a utility of 1).
We will fully characterize the planner’s solution when cP
h > RK , which is relatively simpler than the
market solution (solving for the market solution fully is much more involved). It turns out that when
l<
RK
<h
RC
P
holds, then cP
h > RK always holds. More importantly, this Appendix shows that when ch > RK , then our
key Proposition 6 in the main text remains valid. It is because in Proposition 6 we show the overinvestment
P
result by establishing in the limit that ch ! h < RK . Then, since cP
h > RK , we know that ch < ch in this
case automatically, and the key overinvestment result holds.
C.1.1
Mechanism design approach: constrained e¢ cient allocation at idiosyncratic
stage
We …rst show that the mechanism design approach yields the same result as if the planner opens the trading
market at the idiosyncratic stage: as discussed above, cash …rms ends up with C +R2 K K amount of cash,
while capital …rms have K units of capital but with C R2 K K amount of cash.
Recall that cash …rms cannot operate capital, but capital …rms can consume cash at its reservation value
of 1. Denote the allocations by fCC ; CK ; KC ; KK g 2 R4+ where the subscript indicates the reported type.
Given resource pair (C; K), the planner is maximizing
max
fCC ;CK ;KC ;KK g2R4+
s:t:
1
1
(RC CC ) + (RK KK + CK )
2
2
CC + CK C;
KC + KK K;
CC CK ;
RK KK + CK RK KC + CC :
57
(C.40)
(C.41)
(C.42)
(C.43)
Proposition C.1 The solution to the above problem is
KK = K, KC = 0, CK =
C
RK K
C + RK K
; and CC =
;
2
2
which is identical to the market solution, where the capital price is capped at pb = RK .
Proof. We have several key observations. First, reducing KC and increasing KK until (C.41) binds
can improve objective, relax (C.43), but still satis…es (C.41). Hence KC = 0 and KK = K. Second, (C.40)
holds with equality. Otherwise, let " = (C CC CK ) =2 > 0 and raise CC and CK by ", which improves
objective and satis…es (C.42) and (C.43). Then we guess (C.43) binds before (C.42). Solve the problem with
binding (C.42), we have CK = C R2 K K and CC = C+R2 K K . Since (C.42) holds with strict inequality under
this solution, our claim follows.
C.1.2
Property of the planner’s value function jP (c)
The previous subsection shows that when c > RK , then the aggregate surplus at the idiosyncratic stage is
RK + RC
c + RK
2
+
c
RK
2
where the second (third) term captures the cash in the cash (capital) …rms. For c
RK + cRC . As a result, we can write the HJB equation as
2
0=
2
jP00 (c) +
RK +
RC 1
RC + 1
c+
min (c; RK )
2
2
where the ‡ow payo¤ f (c) RK + RC2+1 c +
in c.
The general solution can be written as
jP (c) =
8
<
:
RC 1
2
+
RC +1
2 c
for c 2 0; cP
h
min (c; RK ) is piecewise linear, increasing, and concave
RK + RC c + D1below e
RC +1
2 RK
jP (c)
RK , the surplus is still
c
+ D2below e
+ D1upper e
c
c
+ D2upper e
c
for c 2 (0; RK )
for c 2 RK ; cP
h
where D1below , D2below , D1upper , and D2upper are coe¢ cients to be determined. Now we list all the boundary
conditions. At cP
l = 0 we have the smooth pasting condition:
jP (0) = ljP0 (0) :
(C.44)
At c = RK we have value matching and smooth pasting conditions on both sides
jP (RK ) = jP (RK +) ; jP0 (RK ) = jP0 (RK +)
(C.45)
At c = cP
h we have smooth pasting condition
P
0
P
jP cP
h = h + ch jP ch
(C.46)
and super contact condition
jP00 cP
h =0
(C.47)
These …ve conditions (C.44)-(C.47) pin down …ve unknowns, in which four of them are coe¢ cients for jP (c)
in two intervals, and one of them is the optimal upper threshold cP
h.
Now we show that our main result hold in this case of cP
>
R
. First, we the counterpart of Proposition
K
h
58
2.
Proposition C.2 The planner’s value function under optimal policy jP (c) is strictly concave, and jP (c) <
f (c) = RK + RC2+1 c + RC2 1 min (c; RK ).
Proof. Denote the ‡ow payo¤ by
f (c) = RK +
RC + 1
RC 1
c+
min (c; RK ) :
2
2
The value function jP (c) satis…es
2
0=
2
jP00 (c) + (f (c)
jP (c))
(C.48)
0
P
00
P
with boundary conditions jP (0) = ljP0 (0) ; jP cP
= h + cP
= 0, and two conditions
h
h jP ch ; jP ch
RC +1
P
P
at c = RK . Note that the boundary conditions imply that jP ch = f ch = 2 RK + RC2+1 cP
h given
cP
h > RK .
We show that jP (c) is concave over 0; cP
h ; which implies jP (c) < f (c). First, from smooth pasting
condition at cP
h we have (recall the parameter restriction of RK > h)
RC + 1
2
jP0 cP
h =
RC +1
2 RK
+ RC2+1 cP
R C + 1 h RK
h
=
< 0:
P
2
h + ch
h + cP
h
jP cP
RC + 1
h
=
P
2
h + ch
RC + 1
2
Then, taking derivative again on (C.48) and evaluate at the optimal policy point cP
h , we have
jP000 cP
h =
2
RC + 1
2
2
jP0 cP
h
=
2 RC + 1 RK h
> 0;
2
2
h + cP
h
(C.49)
and as a result jP00 cP
< 0. Suppose that jP fails to be globally concave over 0; cP
h
h . Then there exists some
P
00
00
00
c) < 0. At
c) = 0 and jP000 (b
c; ch with jP (b
c so that jP is concave over b
point jP > 0, and pick the largest one b
0
0
b
c we have jP (b
c) = f (c), jP (b
c) < f (c) (because jP (c) cross f (c) from above), and jP (c) is strictly convex
around the vicinity of c < b
c. Using standard argument one can show that jP (c) is strictly convex over [0; b
c]
(if not, pick the largest point point e
c so that jP00 (e
c) = 0. But due to convexity of j (c) and linearity of f (c)
c) = 0). Thus jP (c) is strictly convex over
in [e
c; b
c] we have f (c) < jP (c) strictly, contradicting with jP00 (e
c) < f 0 (b
c) < f 0 (0) = RC and jP (0) > f (0) = RK . It
[0; b
c]. Since f (c) is concave, we have jP0 (0) < jP0 (b
contradicts with the boundary condition at c = 0, because jP (0) = ljP0 (0) < lRC < RK .
To sum up, jP (c) is globally concave over 0; cP
h , which also implies that jP (c) < f (c) due to (C.48).
We now show that when RK < hRC holds, we have cP
h > RK .
Corollary 1 Under parameter restriction C.39, and suppose that RK < hRC . Then cP
h > RK :
Proof. Suppose that cP
<
R
;
then
we
should
have
the
same
characterization
in
Proposition
2. However,
K
h
RK
RK
hRC cPh
e
(1 + l )
lRC
(1
l )e
cP
h
2
cP
h +h =0
admits no solution: if RK hRC < 0 then even the …rst term is negative.21 Hence cP
h > RK . And, when
c ! 1 the marginal value of cash is 1 and the marginal value of capital is RK . Hence when RK < h, holding
22
cash is strictly dominated by holding capital when c ! 1, implying cP
h < 1.
21
P
P
P
P
Note that ech (1 + l ) (1 l ) e ch > ech
e ch > 0.
22
This result is also consistent with condition (C.49) which says that postponing liquidating for c > cP
h gives
00
jP cP
h + > 0.
59
Intuitively, when accumulated cash (relative to the capital stock) is below RK , then the marginal value
of cash is RC . If RK < hRC , then the bene…t of capital is below the cost of building capital, it is never
optimal to build the capital for c < RK . When c > RK , the marginal value of cash is just its consumption
value 1. As RK > h says the bene…t of capital exceeds the cost, then the planner starts building capital
P
when c = cP
h > RK for su¢ ciently high ch .
C.1.3
Welfare results
Now we have the counterpart of Proposition 3, which gives the key result about investment ine¢ ciency. The
argument is almost identical to Proposition 3 which only relies on the concavity of jP for all policy c 2 [cl ; ch ]
if ch < cP
h and cl > 0.
Proposition C.3 For any ch < cP
h and cl > 0, we have
@jP (c; cl ; ch )
@jP (c; cl ; ch )
< 0, and
> 0 of all c 2 [cl ; ch ] :
@cl
@ch
P
Proof. Suppose that we are given the policy pair (cl ; ch ) with 0 < cl < ch < cP
h where ch satis…es
P
P
00
the super-contact condition jP ch ; 0; ch = 0: To avoid cumbersome notation we denote the social value
jP (c; cl ; ch ) given the policy pair (cl ; ch ) by j (c; cl ; ch ), and denote the social value under the optimal policy
jP c; 0; cP
h by jP (c). We need to show that
@j (c; cl ; ch )
@j (c; cl ; ch )
< 0 and
> 0.
@cl
@ch
1 1
2 2
This result further implies that for 0 < c2l < c1l < c1h < c2h < cP
h , we have j c; cl ; ch < j c; cl ; ch :
00
00
As preparation, we …rst show that j (ch ; cl ; ch ) < 0 and j (cl ; cl ; ch ) < 0: Because (cl ; ch ) is suboptimal,
2
we must have j (c; cl ; ch ) < jP (c) f (c) (recall the above proposition). Then 0 = 2 j 00 (c) + (f (c) j (c))
implies that j (c) is strictly concave at both ends. Second, for any policy pair (cl ; ch ) (including the market
solution or the planner’s solution), the smooth pasting condition (not optimality condition!) at the regulated
ends implies that
j (ch ; cl ; ch ) (ch + h) j 0 (ch ; cl ; ch )
j (cl ; cl ; ch ) (cl + l) j 0 (cl ; cl ; ch )
=
=
0;
0:
(C.50)
(C.51)
Now we start proving the properties for the top policy ch . De…ne Fh (c; cl ; ch ) @c@h j (c; cl ; ch ), which is
the marginal impact of changing the top investment policy on the social value. Di¤ erentiating the basic ODE
2
@
by the policy ch , we have 2 @c@h j 00 (c; cl ; ch )
@ch j (c; cl ; ch ) = 0, or
2
2
Fh00 (c; cl ; ch )
Fh (c; cl ; ch ) = 0:
(C.52)
Moreover, take the total derivative with respect to ch on the equality (C.50), i.e., take derivative that a¤ ects
both the policy ch and the state c = ch , we have
@
@ 0
j (ch ; cl ; ch ) + j 0 (ch ; cl ; ch ) = j 0 (ch ; cl ; ch ) + (ch + h)
j (ch ; cl ; ch ) + j 00 (ch ; cl ; ch )
@ch
@ch
@ 0
@
)
j (ch ; cl ; ch ) (ch + h)
j (ch ; cl ; ch ) = (ch + h) j 00 (ch ; cl ; ch ) < 0
@ch
@ch
) Fh (ch ; cl ; ch ) (ch + h) Fh0 (ch ; cl ; ch ) < 0:
(C.53)
which gives the boundary condition of Fh ( ) at ch . At cl we can take total derivative with respect to ch on
60
the equality (C.51), we have the boundary condition of Fh ( ) at cl :
@
@ 0
j (cl ; cl ; ch ) = (cl + l)
j (cl ; cl ; ch ) ) Fh (cl ; cl ; ch )
@ch
@ch
(cl + l) Fh0 (cl ; cl ; ch ) = 0:
(C.54)
With the aid of these two boundary conditions, the next lemma shows that Fh ( ) has to be positive always.
Because of the de…nition of Fh (c; cl ; ch ) @c@h j (c; cl ; ch ), it implies that raising ch given any state c and any
lower policy cl improves the social value. The argument for the e¤ ect of cl is similar and thus omitted.
Lemma B.1 We have Fh (c) > 0 for c 2 [cl ; ch ].
Proof. We show this result in three steps.
First, Fh (c) cannot change sign over [cl ; ch ]. Suppose that Fh (cl ) > 0; then from (C.54) we know that
Fh0 (cl ) > 0. Then simple argument based on ODE (C.52) implies that Fh ( ) is convex and always positive.
Now suppose that Fh (cl ) < 0; then the similar argument implies that Fh is concave and negative always.
Finally, suppose that Fh (cl ) = 0 but Fh changes sign at some point. Without loss of generality, there must
exist some point b
c so that Fh0 (b
c) = 0, Fh (b
c) > 0 and Fh00 (b
c) < 0. But this contradicts with the ODE (C.52).
0
Fh (c). As a
Second, de…ne Wh (c) Fh (c) (l + c) Fh (c) so that Wh0 (c) = (l + c) Fh00 (c) = 2 (l+c)
2
result, Wh0 (c) cannot change sign. Because we have Wh (cl ) = 0, Wh (c) cannot change sign either.
Third, suppose counterfactually that Fh (c) < 0 so that Wh0 (c) > 0. Step 2 implies that Wh (c) > 0, and
0
Fh (ch ) = hl+cl (Fh Wh ) < 0. But we then have
Wh (ch ) = Fh (ch )
(l + c) Fh0 (ch ) = Fh (ch )
(h + c) Fh0 (ch ) + (h
l) Fh0 (ch ) < 0;
where we have used (C.53), contradiction. Thus we have shown that Fh (c) > 0.
The next proposition naturally follows.
Proposition C.4 When ! 1 then in the market solution ch ! h < RK . For planner’s solution, we
have cP
hRC > 0 is su¢ ciently small or when RK hRC < 0. It implies that …rms
h > RK when either RK
overinvest in capital in booms in the market solution.
C.1.4
A Model without Idiosyncratic Investment Opportunities
Suppose that at the …nal date, every …rm with holdings (K; C) can produce RK K + RC C units of …nal
consumption goods. This formulation is also equivalent to the base model but with complete market, i.e.,
introducing some Arrow-Debrew securities contingent on …rms’ idiosyncratic type realization–either K or
C— which obviously complete the market. Ex ante, each …rm will fully hedge using these Arrow-Debrew
contracts, so that each unit of capital pays o¤ RK units of consumption goods while each unit of cash pays
o¤ RC units of consumption goods. This is exactly identical to the hypothetical precautionary-saving-motive
model without idiosyncratic investment opportunities.
We show that the precautionary-saving-motive model is constraint e¢ cient, a reminiscent of the …rst
welfare theorem. To prove this result formally, denote the value functions in the complete market equilibrium
by vcm (c) and qcm (c). The HJB equation for value functions become
2
0
=
00
qcm
(c) + (RC qcm (c))
|
{z
}
|2 {z }
volatility of dct
2
0
=
00
vcm
(c) +
2
| {z }
volatility of dct
(C.55)
…nal date realization
0
qcm
(c) 2
| {z }
exp ected value of dividends
61
+ (RK vcm (c))
|
{z
}
…nal date realization
(C.56)
In contrast, in our model the valuation equations for q and v given idiosyncratic investment opportunities
are
2
0
00
qcm
(c) + (RC qcm (c)) +
|2
{z
} |2
|2 {z }
=
volatility of dct
2
0
00
vcm
(c) +
|2 {z }
=
volatility of dct
b ecom ing a cash …rm
RK
c
(C.57)
qcm (c)
{z
}
b ecom ing a capital …rm
0
qcm
(c) 2
| {z }
exp ected value of dividends
+
|2
(RC c
{z
vcm (c)) +
(RK vcm (c))
}
|2
{z
}
b ecom ing a cash …rm
(C.58)
b ecom ing a capital …rm
Take q equation as example. In (C.55), the term "…nal date realization" captures that with intensity , the
…rm can use its cash holding to obtain RC units of consumption goods. While in (C.57), this term has two
components: if becoming a cash …rm with intensity 2 , then it obtains RC ; while if becoming a capital …rm
with intensity 2 , it obtains RpbK = RcK by purchasing capital (which generates RK ) at the price of pb = c.
Comparing (C.55)-(C.56) to the planner’s solution in Section 3.1.1, we see that
jP (c) = vcm (c) + cqcm (c) :
Denote the new endogenous (dis)investment boundaries by ccm
and ccm
h
l . Because we knew that the concm
strained e¢ cient solution features cl = 0, we need to be careful in the associated boundary conditions at
the lower bound boundary. At upper boundary, we have
v (ccm
0 cm
h )
= h, v 0 (ccm
h ) = 0, and q (ch ) = 0:
q (ccm
h )
(C.59)
At the lower boundary, taking into account of possibility that cl = 0 binds at zero, we have the complementaryslack condition:
v (ccm
0 cm
l )
= l, v 0 (ccm
l ) = 0, and q (cl )
q (ccm
l )
The …rst condition
ccm
l ,
v(cl )
q (cl )
0 with strict inequality if ccm
= 0:
l
(C.60)
= l have to hold because in equilibrium only a fraction of …rms are liquidating their
capital at
who must be indi¤erent between selling or liquidating their capital. The intuition behind
= 0) < 0 is as follows. Note that c in the functions v ( ) and q ( ) capture the
)
=
0
and
q 0 (ccm
v 0 (ccm
l
l
aggregate cash liquidity. Loosely speaking, if the aggregate liquidity is strictly negative say ", then the
value of cash is higher (relative to q (c = 0)) because cash can be used to reduce the amount of capital that
needs to be liquidated. In contrast, given c = " or 0, the optimal policy with regard to existing capital is
unchanged (think about those capitals that end up not to be liquidated), which explains v 0 (ccm
= 0) = 0.
l
cm
P
The following proposition formally shows that ccm
=
0
and
c
=
c
,
which
coincide
with
the
planner’s
l
h
h
solution.
Proposition C.5 In the complete market economy, there is an equilibrium for any set of parameters where
1. …rms do not consume before the aggregate stage,
2. each …rm in each state c 2 [0; ccm
h ] is indi¤ erent in the composition of her portfolio,
3. each …rm holding capital use every positive cash shock to build capital if and only if c = ccm
h and …nance
the negative cash shocks by liquidating the capital if and only if c = 0;
4. the value of holding a unit of cash and the value of holding a unit of capital are described by
qcm (c) = RC + e
vcm (c) = RK + e
c
c
L1 + ec L2 ;
D1P cL1 + ec
62
D2P
cL2 :
(C.61)
(C.62)
where constants L1 ; L2 ; D1P ; D2P and ccm
are determined by boundary conditions from (C.59) and
h
(C.60):
vcm (ccm
vcm (0)
0
0
cm
0
h )
= h;
= l, vcm
(ccm
(C.63)
h ) = qcm (ch ) = vcm (0) = 0:
qcm (ccm
)
qcm (0)
h
In this equilibrium, vcm (c) is increasing in c; qcm (c) is decreasing in c. Hence pcm (c)
increasing in c, implying the optimality of ccm
= 0,
l
vcm (c)
qcm (c)
is
P
P
P
5. …nally, we have ccm
h = ch , D1 = D1 and D2 = D2 given in the planner’s solution, so that jP (c) =
P
vcm (c) + cqcm (c) for all c 2 0; ch .
Proof. Given (C.61)-(C.62), the boundary conditions in (C.63) are
cm
cm
ech
D2P
cm
ccm
h L2
cm
ch
RK + RC ccm
D2P + e ch D1P
h +e
cm
cm
c
RC + e h L1 + ech L2
RK + D2P + D1P
RC + L1 + L2
cm
ccm
P
cm
h
e
D1
ch L1
L1 e ch
L2 ech
ccm
h
e
D2P
ccm
h
L1 + e
L2
D1P
L2
L1
= h + ccm
h ;
(C.64)
= l;
(C.65)
=
0;
(C.66)
= 0;
= 0:
(C.67)
(C.68)
Adding ccm
h times (C.67) to (C.66) gives
cm
ech
D2P
L2
ccm
h
e
D1P + L1 = 0:
Together with (C.68), this implies
D2P = L2 ; and
Substituting this into (C.67) gives
e
Also, as L1 + L2 = L3
ccm
h
L1 = D1P :
(C.69)
D2P = 0:
(C.70)
cm
D1P + ech
L4 , (C.65) implies that
RK + D2P + D1P = l RC + D2P
D1P
(C.71)
and by (C.69), (C.64) is equivalent to
cm
ch
RK + RC ccm
h +e
D2P + e
ccm
h
D1P = (h + ccm
h ) RC
D1P e
ccm
h
cm
+ D2P ech
:
(C.72)
Then, we observe that the system (C.70)-(C.72) is equivalent with the following system for the planner’s
P
problem with D2P = D2 , D1P = D1 and ccm
h = ch :
RK + D1 + D2
RK + RC cP
h + D1 e
D1 e
with D1 =
P
lRC )e2 ch
2 cP
h
)e
(1 l
(RK
(1+l
)
cP
h
cP
h
, D2 =
+ D2 e
+ D2 e
= l (RC
cP
h
=
cP
h
=
RK
(1+l )e2
h + cP
h
RC
D1 e
cP
h
+ D2 e
cP
h
;
0:
lRC
cP
h
D 1 + D2 ) ;
(1 l )
.
0
Finally, to show that price is monotonically increasing in this economy we show that vcm
(c) > 0 and
63
0
qcm
(c) < 0 for every c 2 (0; ccm
h ). It is easy to check that
0
qcm
(c)
=
c
e
2
=
L1 + ec L2 =
lRC ) ec
(RK
P
e2 ch
2
c
e
1
D1 +
P
e2 (ch
+l
e2
2 c
e D2 =
c)
< 0:
cP
h
1 +1
This also veri…es the complementarity-slackness condition in (C.60). And,
0
vcm
(c)
=
ec (D2
=
2
c
L2 ec
cL2 )
c
D2 e
c
2
D1 e
c
c
e
2
=c
(D1
(RK
cL1 )
L1 e
P
e2 (ch
c
lRC ) e
c
P
e2 ch
+l
c)
P
e2 ch
1
0:
1 +1
Q.E.D.
C.2
The role of Cobb-Douglas technology in Section 5.1
In Section 5.1 we introduce a Cobb-Douglas technology which combines cash and capital to produce some
…nal goods; and recall in the base model this technology is not needed. This note explains why we introducing
this technology.
It is a tradition in the pecuniary externality literature to think about distortionary tax scheme as small
transfer which results in some …rst-order incentive/welfare implications. But, because the linear production
technology used in the main model may well push individual …rms to take some cornered solution (as we
…nd in our numerical solution), …rms are insensitive to marginal tax transfers. In other words, the linear
technology implies a cornered solution on the lower side, and as a result distortionary tax schemes which
only a¤ect incentives slightly have a hard time to induce some real e¤ect.
We introduce the Cobb-Douglas technology K C 1 in the aggregate stage in order to break the corner
solution for the disinvestment threshold. Cobb-Douglas technology naturally implies a higher marginal value
of cash when the aggregate cash level is lower. This can be seen here:
0
= q 0 (c)
2
+ v 0 (c)
p (c) +
2
c2
p (c)
2
+
2
v 00 (c) +
RC p (c) + RK
2
v (c) +
(C.73)
c1
| {z }
extra mv of capital
0
= q 0 (c)
2
2
p (c) +
c
p (c)
2
+
2
q 00 (c) +
1
2
RC +
RK
p (c)
q (c) + (1
)c
|
{z
}
(C.74)
ex mv of cash
relative to (12), there is an extra ‡ow (1
) c in the cash value equation which increases when c drops.
In fact, because it features the Inada condition so that the marginal value of cash goes to in…nity when
Ct = 0, it automatically guarantees that the equilibrium disinvestment threshold take an interior solution
with a zero-order …rst condition (rather than be cornered at c = 0 with non-zero …rst-order condition).
As a result, with Cobb-Douglas technology, in the market solution …rms are taking interior solutions
on both the upper investment and lower disinvestment thresholds, both with zero …rst-order conditions.
Because policy interventions are in the form of small distortionary tax schemes, this helps us illustrate the
pecuniary externality with small policy interventions, an exercise that we are performing in Section 5.
C.3
The model with collateralized borrowing and proof of Proposition 9
As a preparation we …rst analyze the model with collateralized borrowing; we then give the proof of Proposition 9 in Section 5.2.
We highlight three parameters with superscript b which have special roles in characterizing the equilibb
rium. In the economy with collateralized borrowing, we use RK
to denote the productivity of capital; and,
64
per unit of capital …rms get lb units of cash when liquidating while need to invest hb units of cash when
investing.
We conjecture that …rms always max out their borrowing capacity (and verify later this holds in equilibrium). In the idiosyncratic stage after the heterogenous technology shocks hit, there are K =2 units of
capital to be sold. On the cash side, in addition to C =2 units of the cash, collateralized borrowing implies
that there are bK units of extra cash in aggregate from external creditors. Hence the equilibrium capital
=2
= 2b + c : For capital …rms being willing to purchase capital, we require
price is pb = bKK+C
=2
b
RK
> 2b + c ;
(C.75)
which, as in the base model, can hold in equilibrium because c will be bounded endogenously.
For assets that can be used as collateral, it is useful to note that pb
b is the “e¤ective” capital price.
This is because each unit of capital can be used to borrow b units of cash, and …rms with their own cash of
pb
b = b + c can buy a unit of capital. When c = b which is lower bound of the aggregate net cash
holding, the e¤ective price of capital drops to zero.
For capital …rms with capital K and cash C , they will use their cash holding C , together with credit
K b, to purchase capital from the market at the e¤ective capital price b + c . The …nal consumption goods,
net of borrowing payment, is
K
|
b
RK
{z
b
C +K b
b
RK
b+c
|
{z
+
}
own capital holdings
=K
b
}
purchasing capital from m arket
b
(2b + c ) RK
b+c
{z
|
b
+C
}
m arginal payo¤ of capital
b
b
RK
b+c
| {z }
(C.76)
m arginal payo¤ of cash
Note, under condition (C.75), it is optimal to exhaust the borrowing capacity (at a marginal cost of 1) to
puchase the capital from the market (at a marginal bene…t of
+
C RC
| {z }
net cash holdings
b
RK
b
b+c ).
For cash …rms, their payo¤ is
K (2b + c ) RC
|
{z
}
:
(C.77)
selling capital to the m arket
Hence, the marginal payo¤ for capital is K (2b + c ), while for cash it is RC .
Now we move on to aggregate stage, and denote the value of capital and cash by v b (c) and q b (c),
respectively. The similar structure for the market equilibrium as in the base model prevails, i.e. …rms build
(dismantle) capital when the aggregate cash-to-capital ratio ct reaches an endogenous upper (lower) threshold
chb (cl b ). In the inaction region cb 2 cbl ; chb , we have the same evolution of state variable dct = dZt , and
the values of capital and cash satisfy
b0
0
= q (c)
0
=
2
2
2
2
+
q b00 (c)
2
v
b00
b
(2b + c) RK
v (c) +
2
b+c
(c)
q b (c) +
"
b
RC +
2
b
#
+ RC (c + 2b) ;
b
RK
b
:
b+c
Here, we have used the marginal payo¤s of capital and cash for either capital or cash …rms given in (C.76)
and (C.77). The boundary conditions are the same as the base model:
q b0 cl b
q
b0
chb
= v b0 cl b = 0, v b0 cl b = lb q b cl b ;
= v
b0
chb
= 0, v
b
chb
b b
=h q
chb
:
(C.78)
(C.79)
We now show that there is a simple relationship between the economy with and without borrowing both
65
in the planner’s case and in the decentralized case. de…ne three “translated” parameters as
b
RK = RK
b, h = hb
b, and l = lb
b;
(C.80)
and consider the no-borrowing economy with the above three parameters. For the market equilibrium, denote
the resulting capital and cash value functions v (c) and q (c) respectively, with equilibrium thresholds (cl ; ch ).
Analogously, denote j ( ) and cP
h as the social planner’s solution. We have the following proposition.
Proposition C.6 Consider the economy with borrowing. For the social planner, we have j b (c) = j (c + b),
P
cP;b
= b and cP;b
b. For the market equilibrium the capital and cash value functions are given by
l
h = ch
v b (c) = v (c + b) + bq (c + b) ; and q b (c) = q (c + b) :
(C.81)
Hence the capital price is pb (c) = p (c + b) + b, and the investment and disinvestment thresholds are given
by cbl = cl b and cbh = ch b:
Proof. Recall that in the base model without borrowing, if R = Rb b, our value functions satisfy
2
q 00 (c) +
0
=
0
= q 0 (c)
2
2
2
2
+
2
(RC
q (c)) +
v 00 (c) +
2
(Rc c
2
b
b
RK
c
v (c)) +
q (c) ;
2
Rb
b
(C.82)
v (c) :
(C.83)
We only show q b . If q b (c) = q (c + b), we have
2
0=
2
2
, 0=
2
q b00 (c) +
2
q 00 (c + b) +
q b (c) +
RC
(RC
2
2
b
RK
b
c+b
q (c + b)) +
2
q b (c)
b
RK
b
c+b
q (c + b)
which holds always as we can view c + b as c in (C.82). Similarly we can show the result for v b (c). The
investment and disinvestment thresholds and the social planner’s solution are obvious given this result.
C.3.1
Proof of Proposition 9
For simplicity we set
= 1. Our results rely on two lemmas. The …rst lemma gives the market solution.
Lemma B.2 For any a > 1; cl = x; ch = ax is an equilibrium in the limit x ! 0; if
3 (a
3 (a
1)
(a + 1) ln a
x = l;
ln a
1) (a + 1) ln (a)
x + o (x) = h.
ln (a)
Proof. One can show limx!0 xfl (x; ax) =
RK +
pl (x; x; ax)
= lim
x!0
x!0
x
lim
RK +
=
lim
x!0
RK
2
x R2C
1
+
a ln(a)
a 1
ln(a4 )
RK 4(a
1)
ln(a)
4(a 1) ; lim"!0 gl
xRC
2
+
(x; ax) = 1
a ln(a)
a 1 .
RC
RK
2 m (x; ax) + 2 (gl (x; ax)
x R2C + RK xfl (x; ax)
ln(a4 )
4(a 1)
=
66
3 (a
1)
(a + 1) ln a
:
ln a
Thus, we have
xfl (x; ax))
Hence if l =
3(a 1) (a+1) ln a
x
ln a
then x = L (ax) in the limit. Similarly
lim axfh (x; ax) =
x!0
a ln (a)
; lim gh (x; ax) = 1
(a 1) x!0
ln (a)
:
a 1
Thus, for
ph (x; x; ax)
x!0
x
lim
=
=
=
lim
RK +
axRC
2
+
RC
RK
2 m (x; ax) + 2 (gh (x; ax)
x R2C + RK xfh (x; ax)
RK +
axRC
2
+
RC
2 m (x; ax)
x!0
lim
RK
2
1
ln(a)
a 1
=
a ln(a)
a 1
x R2C + RK xfh (x; ax)
x!0
1+
+
axfh (x; ax))
1
2
(gh (x; ax) axfh (x; ax))
3 (a
=
1
2 xfh (x; ax)
1)
(a + 1) ln (a)
ln (a)
Hence if
h=
3 (a
1)
(a + 1) ln (a)
x + o (x)
ln (a)
Then x = H (ax) in the limit. QED.
Because l (k) = O "k , h (k) = O "k + " and k < 1, " = o O "k , which implies the above lemma
applies. Hence the market solution has
cl = O "k , ch = aO "k
where a > 1 is the solution to 3 (a
1) = a ln a. The next lemma gives the planner’s solution.
Lemma B.3 For x being su¢ ciently small, suppose that
l = x, and h = x + O x3(1+
where the constant
can be either positive or negative. Then the social planner’s solution satis…es
cP
h
/
Proof. Without loss of generality we …x
RK
)
x + O x3(1+
)
RC
h
P
8
>
>
>
<
x1+
x
>
>
>
: x1+1:5
ech (1 + x)
if
<0
if
=0 :
if
>0
= 1. The planner’s solution cP
h satis…es
(1
x) e
67
cP
h
i
= 2 (RK
3(1+
xRC ) cP
h +x+O x
)
(C.84)
It is easy to show cP
h ! 0 as x ! 0. Then Taylor expansion implies that
P
ech (1 + x)
1 + cP
h +
=
(1
1
2
x) 1
2x + 2cP
h +x
=
P
x) e ch
1 P 3
2
3
cP
+
c
+ o cP
(1 + x)
h
h
6 h
1 P 2 1 P 3
3
cP
c
c
+ o cP
h +
h
2 h
6 h
1
2
3
3
cP
+ o cP
cP
+
h
h
h
3
(1
hence the RHS of (C.84) is
x + O x3(1+
RK
=
)
P
2RK x + 2RK cP
h + RK x ch
RC x + O x3(1+
)
P
2x + 2cP
h + x ch
RC
RK P 3
ch
3
RC P
2
ch
3
2
+
1
3
cP
h
3
3
x + O x3(1+
)
)
+o
3
cP
h
+o
2 RC x + O x3(1+
+
x cP
h
2
x
3(1+
2RC cP
h x+O x
cP
h
3
)
;
the LHS of (C.84) is
3(1+
2RcP
h + 2RK x + 2RK O x
)
2RC xcP
h
2RC x x + O x3(1+
)
;
and (C.84) therefore is
2
RK x cP
h
RC P
ch
3
=
+
3
RK P
ch
3
3
3(1+
2RC cP
hO x
x + O x3(1+
)
cP
h
+o
3
)
RC x + O x3(1+
+ 2RK O x3(1+
)
)
x cP
h
2
:
Now, we write the above equation as
RK P 3
2
+
ch
RK x cP
h
| {z } | 3 {z }
(a)
=
x cP
h
|
(b)
2
2RK O x3(1+
|
{z
(c)
RC x + O x3(1+
{z
)
(1)
)
}
RC P
3(1+
c + cP
hO x
3 h
} |
)
1 P
RC 2 +
c
3 h
{z
(2)
2
}
o
|
cP
h
{z
(3)
3
}
Here, term (1) on LHS is dominated by (a) on RHS, term (2) on LHS is dominated by (c) on RHS, and the
term (3) on LHS is dominated by (b) on RHS. As a result, it must be that cP
h is determined by LHS=0 when
x is su¢ ciently small. We have the following three cases to consider.
> 0, we conjecture that term (b) is at a higher order so it is negligible. Thus cP
h is determined by
1
p
p
2
P 2
3(1+ )
P
3(1+ )
1+1:5
x ch RK 2RK O x
= 0, or ch = 2 O x
=x = 2O x
. This also implies
1. If
that cP
h
2. If
3
= O x3+4:5
which is indeed at a higher order than term (c).
= 0, then we LHS=0 implies that
2RK O "3
RK P
ch
3
68
3
= RK x cP
h
2
which implies that cP
h = O (x).
3. If
< 0, we conjecture that term (a) is at a higher order so it is negligible. Then cP
h is determined by
RK
3
which implies that cP
h =
p
3
1+
6 (x)
cP
h
3
2RK O x3(1+
)
=0
.
Recall that l (k) = O "k , h (k) = O "k + "; applying the above lemma, we know that
cP
h (k) /
8
>
>
>
<
O "1=3
O "k
>
>
>
: O "(1 k)=2
if k < 1=3
if k = 1=3
if k > 1=3
Because cl (k) = O "k and ch (k) = aO "k , for k > 1=3 we have cP
h (k) > ch (k), i.e. overinvestment in
booms.
C.4
An alternative equilibrium
In the main text we showed that an equilibrium exist when h l is su¢ ciently small; it is possible that the
type of equilibrium presented in the main text does not exist. In this subsection we provide some insights
on the type of equilibrium that arises instead.
While the system (12)-(13), (7)-(9) always have a solution, for some parameters this solution implies that
for a c su¢ ciently close to cl ; the price is below the threshold l: This obviously cannot be an equilibrium—
…rms would dismantle whenever the price drops below the liquidation bene…t l. For that set of parameters
we can construct the equilibrium as follows. There exists a cx 2 (cl ; ch ), so that for every c 2 [cl ; cx ] we
have p (c) = v(c)
q(c) = l, and an endogenous fraction of capital are dismantled at every instant. That is, in this
range the price is constant in c and …rms dismantle an increasing fraction of their capital as c drops further
from cx . The following proposition describes this equilibrium.
Proposition C.7 Suppose that there is a ch < RK ; cx 2 (l; ch ) ; q0 ; A1 ; A2 ; A3 ; A4 solving (12)-(13), (31)
RK
(l cx ) = q 0 (cx )
cx
RK
l 2 RC +
(l cx ) = v 0 (cx )
2
cx
v (ch )
= l;
= h; v 0 (ch ) = q 0 (ch ) = 0:
q (ch )
2
v (cx )
q (cx )
2
RC +
Then there exists a market equilibrium with partial liquidation where
1. …rms do not consume before the …nal date,
2. each …rm in each state c 2 [l; ch ] is indi¤ erent in the composition of its asset holdings,
3. …rms do not build or dismantle capital when c 2 (cx ; ch ) and, in aggregate, …rms spend every positive
cash shock to build capital i¤ c = ch and cover the negative cash shocks by liquidating a fraction of
capital i¤ c 2 [l; cx ] : When c = l; …rms …nance every negative cash shock by liquidating capital.
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4. the value of cash and the value of capital are given by q (c) and v (c) are the same as in the base model
for c 2 (cx ; ch ); for c 2 (l; cx ) they are
q (c) = q0 +
2
(RC l
2
RK ) (c
RC 2
c
2
l)
l2 + lRK (ln c
ln l) ; v (c) = lq (c)
and the price in the aggregate stage is p = l when c 2 [l; cx ].
5. In the idiosyncratic stage, each cash …rm sells all its capital to the cash …rms who are not hit by the
shock of the price p^ = c:
Proof. Under the conditions of the Proposition, …rms start to disinvest whenever p (c) = l. Given the
liquidation rate y (c) dt = dK=K, then its impact on the aggregate cash-to-capital ratio c is
dC
K
x (c) dt
C dK
=
K K
ldK
K
C dK
= (l + c) y (c) dt;
K K
so c evolves as dc = x (c) dt + dZt : We must have v (c) = lq (c) as …rms are always indi¤erent in liquidating
the capital, and v and q satis…es:
2
q 00 (c) +
0
= x (c) q 0 (c) +
0
= x (c) v 0 (c) + q 0 (c)
2
2
RC +
2
2
+
RK
c
v 00 (c) +
2
2
q (c)
(RC c + RK )
v (c)
Using v (c) = lq (c), we obtain
2
0
= x (c) lq 0 (c) +
0
= x (c) lq 0 (c) + q 0 (c)
2
lq 00 (c) +
2
l
2
2
+
2
RC +
lq 00 (c) +
RK
c
2
lq (c)
(RC c + RK )
lq (c)
Eliminating identical terms, we get
q 0 (c) =
2
2
RK
c
RC +
(l
c) = 0:
As q 0 (cl ) = 0 has to hold, cl = l: The closed-form solution is
q (c) = q0 +
2
2
(RC l
RK ) (c
RC 2
c
2
l)
l2 + lRK (ln c
ln l)
And, we have q 00 (c) = 2 2 RC + lRc2K < 0: We know that of c 2 [l; cx ] we have v (c) = lq (c) which allows
us to back out the endogenous drift of c:
2
x (c) =
2
q 00 (c)
2
RC +
q 0 (c)
RK
c
+ q (c)
;
and thus the endogenous liquidation rate y (c) = x(c)
l+c . For c > cx we have the ODE as usual. We then search
of the cx ; ch pair that satis…es the conditions of the proposition.
Plotting v, q and p give very similar graphs to Figure 2 with the main di¤erence that at the range
c 2 [l; cx ] the price is ‡at at the level l: In the same range q (c) is decreasing implying that v (c) = lq (c) is
also decreasing.
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