Fiscal Risk and the Portfolio of Government Programs
Samuel G. Hanson
David S. Scharfstein
Adi Sunderam
Harvard University
June 2014
Abstract
This paper proposes a new approach to social cost-bene…t analysis using a model in
which a benevolent government chooses risky projects in the presence of market failures
and tax distortions. The government internalizes market failures and therefore perceives
project payo¤s di¤erently than do individual private actors. This gives it a “social risk
management” motive – projects that generate social bene…ts are attractive, particularly
if those bene…ts are realized in bad economic states. However, because of tax distortions,
government …nancing is costly, creating a “…scal risk management” motive. Government
projects that require large tax-…nanced outlays are unattractive, particularly if those
outlays tend to occur in bad economic times. At the optimum, the government trades o¤
its social and …scal risk management motives. Frictions in government …nancing create
interdependence between two otherwise unrelated government projects. The …scal risk of
a project depends on how its …scal costs covary with the …scal costs of the government’s
overall portfolio of projects. This interdependence means that individual projects cannot
be evaluated in isolation.
We thank John Campbell, Eduardo Davilla, Matt Weinzierl, and seminar participants at Harvard and
Princeton for helpful comments. We gratefully acknowledge funding from the Harvard Business School Division
of Research.
1
Introduction
In modern economies, a signi…cant fraction of economy-wide risk is borne indirectly by taxpayers via the government. Governments have signi…cant liabilities associated with retirement
bene…ts, unemployment insurance, and …nancial system backstops. These liabilities are large—
the amount of credit risk explicitly recognized on the balance sheet of the US federal government
now exceeds $3 trillion, and implicit or o¤-balance sheet liabilities are even larger. For instance,
o¤-balance sheet guarantees on mortgage-backed securities account for another $5 trillion. And,
during the …nancial crisis, total o¤-balance sheet …nancial system backstops temporarily reached
more than $6 trillion (Geithner, 2014). Moreover, the government’s overall risk is not idiosyncratic but varies systematically with economic conditions. For example, the US debt-to-GDP
ratio rose from 38% to 72% between 2007 and 2013 due to falling tax revenue and increasing
expenditures on government programs that automatically expand in a recession.
Given the sheer magnitude of these exposures, it is critical to understand how the government should choose, price, and manage the set of risks it bears. The standard approach to
government cost-bene…t analysis essentially ignores these issues. Common practice in government budget accounting— enshrined in U.S. budgetary law under the Federal Credit Reform Act
of 1990 (FCRA)— is to determine whether individual programs in isolation have positive net
present values, when their costs are discounted at the government’s borrowing rate, essentially
the riskless rate.
This common practice is incorrect for two reasons. First, the cost of capital for a project
should depend on the cash ‡ow attributes of the project— the outlays it requires and the payo¤s
it generates. Second, since taxation is distortionary, it is costly for the government to raise
…nancing through taxes. Vast literatures in public economics study each of these factors in
isolation. There is a large literature studying the outlays and bene…ts of government programs
like unemployment insurance and social security. And an equally large literature studies optimal
government …nancing holding …xed the set of programs the government wishes to undertake.
In this paper, we bridge the gap between these two literatures, studying the ways in which
distortionary government …nancing interacts with the set of projects a benevolent government
should choose to undertake. The result is a ‡exible framework for conducting social cost-bene…t
analysis in a stochastic environment with distortionary taxation that provides novel implications
for government risk management and risk pricing.
Our setup di¤ers from the Ricardian framework, where the government is simply a veil for the
taxpayer, in two important ways. First, we assume that government taxation is distortionary,
so that a dollar of government revenue costs society more than a dollar. One can think of
this as a short-hand for a variety of frictions associated with government involvement in the
1
private economy. Second, we assume that government projects can generate social bene…ts
that private actors are not able to generate on their own. We model these social bene…ts in
reduced form, but think of them as arising from the fact that the government often has unique
technologies for addressing market failures. For instance, the government may be able to use
regulations to correct externalities, mandate contributions to address public goods problems,
or mandate purchases to address market-unravelling. In this setting, we can characterize the
social welfare maximizing scale of a government program, taking the payo¤s from the project as
given. Equivalently, we characterize the optimal pricing (i.e., cost of capital) for the program,
taking its scale as given.
In the …nance literature, corporate hedging and risk management activities can enhance
…rm value if it is costly for …rms to raise …nancing from external investors (Froot, Scharfstein,
and Stein, 1993). By the same logic, …nancing frictions have implications for the scale and
composition of optimal …rm investment (Bolton, Chen, and Wang, 2011; Bolton, Chen, and
Wang, 2013; Campello, Graham, and Harvey, 2011; Fazzari, Hubbard, and Petersen, 1988;
Kaplan and Zingales, 1997). In our setting, the distortionary costs of taxation play a similar
role, making it costly for the government to raise funds from taxpayers through taxation. The
distortionary costs of taxation induce the government to behave as though it is more risk-averse
than the taxpayers it represents, just like …nancing frictions can make …rms behave as though
they are more risk-averse than shareholders.
In our model, the government has two motives for risk management: a “social risk management”motive and a “…scal risk management motive”. The government’s social risk management
motive arises from the fact that the bene…ts of a government program— the additional output
created from mitigating market failures— may vary cyclically. For instance, the …nancial stability bene…ts of deposit insurance may largely accrue from preventing large-scale bank runs in
recessions, making it a valuable social hedge. Similarly, the bene…ts from reducing labor market
frictions or from …scal stimulus may largely accrue during recessions.
The government’s …scal risk management motive arises from the desire to avoid costly tax
distortions. By taking on projects that are tax hedges, the government can reduce outlays when
de…cits would otherwise be high, thus reducing average tax distortions. Moreover, since the
distortionary costs of government …nancing depend on the total scale of government outlays,
the …scal risk management motive generates interdependence between the optimal scale of two
otherwise unrelated government projects. The intuition for this interdependence corresponds
closely to classic precepts from the theory of portfolio choice (Markowitz, 1952; Tobin, 1958;
Sharpe, 1964; Linter, 1965). Speci…cally, the …scal risk of a program depends on how its …scal
costs covary with the …scal costs of the total government portfolio. This means that costs of
2
individual programs cannot be evaluated in isolation. And when the government …scal burden
is elevated, the cash-‡ow pro…le of a given project in relation to the rest of the government
portfolio is critical.
A key issue we highlight is the interaction between the government’s social and …scal risk
management motives. These motives may con‡ict because programs like deposit insurance
that have strong social risk management bene…ts tend to involve government expenditures and
hence higher tax distortions in bad times, creating more …scal risk. On the other hand, the two
risk management motives may reinforce each other if social risk management preserves private
output in a downturn— reducing the distortionary impact of a given dollar amount of taxes.
We illustrate the novelty of our framework by considering an example in which the government chooses between a “…scally safe” ex-ante technology and a “…scally risky” ex-post
technology for correcting a market failure. For concreteness, we consider the context of interventions aimed at promoting …nancial stability. Regulation of …nancial intermediaries is an
ex-ante technology for promoting …nancial stability: it is …scally safe in the sense that government expenditures on regulation vary little across states of the world. By contrast, capital
injections or lender of last resort operations (i.e., “bailouts”) are ex-post technologies. They are
…scally risky because government expenditures associated with them vary signi…cantly across
states. In our framework, it is optimal for the government to pursue some regulation and some
ex-post bailouts. A key insight is that the relative attractiveness of these two technologies for
producing …nancial stability depends on the total …scal burden of the government and how it
varies across states. When the government’s …scal burden is low, ex-post bailouts are a relatively
attractive way to provide …nancial stability. However, as the …scal burden rises, it is optimal to
substitute toward the less …scally risky technology, namely, regulation. This corresponds to the
classic portfolio choice intuition that investors who face higher level of “background risk”should
choose more conservative …nancial portfolios (Merton, 1973; Campbell and Viceira, 2003). In
addition, we show that when the distortionary costs of taxation rise, the optimal quantity of
…scally risky ex-post bailouts falls. This corresponds to the standard …nance intuition that the
optimal portfolio allocation to risky assets falls as risk aversion rises.
Our paper sits at the intersection of public …nance, corporate …nance, and asset pricing.
Work in public …nance often considers individual government interventions in isolation. There
is, for instance, one strand of literature studying the market failures that justify the use of
unemployment insurance and a separate strand studying the market failures that justify the
use of deposit insurance. However, in a non-Ricardian world, these two problems are not
separable. Our portfolio approach to government …nance tells us that the cost of capital should
be higher for a program with larger expenditures in bad …scal times, when spending on other
3
programs is already high and taxes are therefore elevated.
Work in macroeconomics recognizes that government expenditures may be cyclical and,
assuming that tax-smoothing is imperfect, that the tax burden will therefore be cyclical as
well. However, this work typically treats the government as an exogenously given collection of
programs. By contrast, our approach shows that the cyclicality of government expenditures has
important implications for the set of programs that should be undertaken by the government.
The plan for the paper is as follows. In Section 2 we develop the general model and characterize the optimal scale of a single potentially welfare-improving government program in the
presence of distortionary taxation. Section 3 explores several special cases of the general model
that help clarify the key intuitions developed in Section 2. Section 4 extends the model to
consider portfolios of multiple government programs. In particular, we explore the choice between a …scally safe and a …scally risky program for addressing a given market failure. Section
5 considers a variety of additional extensions of the basic model, and Section 6 concludes.
2
The General Model
2.1
Setup
Exogenous private income is Yt . By correcting some market failure, a government project
or program may generate a social payo¤ of Wt in the form of additional private income. For
instance, if Wt = 0, there are no social bene…ts associated with the project. In contrast, deposit
insurance may create broad …nancial stability bene…ts, so it would have Wt > 0. Suppose the
government does some quantity q of this program. We assume this changes private income to
Yt + qWt and requires government outlays of qXt . We assume that the government can issue
(j)
default-free bonds of any maturity. Speci…cally, let Dt
denote the amount of j-period zero
(j)
coupon bonds issued at time t and due at t + j. Similarly, let Rt
denote the gross j-period
interest rate on these bonds.
The government’s ‡ow budget constraint is given by
Tt +
P1
j=1
(j)
Dt =
P1
(j) j (j)
j=1 (Rt j ) Dt j
+ Gt + qXt ,
(1)
where Tt is taxes, Gt is exogenous government expenditure, and qXt is the endogenous level
of expenditure associated with the program under consideration. Equation (1) says that the
government’s source of funds from taxes and the proceeds from new debt issues must equal its
uses of funds for maturing debt and spending.1
1
Note that Wt must be interpreted as the incremental payo¤ relative to other government projects embedded
4
We assume that taxation is distortionary. Thus, the household budget constraint equating
household expenditures with household income is given by
Ct + Tt +
P1
j=1
(j)
Dt = Yt + qWt
h (Tt ) +
P1
(j) j (j)
j=1 (Rt j ) Dt j ,
(2)
where Ct is private consumption. Here h ( ) represents the distortionary costs or excess burden
of taxation with h0 ( ) > 0 and h00 ( ) > 0:We will frequently choose h (Tt ) = ( =2) Tt2 for
simplicity. Note that in the presence of distortions, we must separately keep track of outlays Xt
and bene…ts Wt rather than just net bene…ts Wt
Household utility is
U = E0 [
where 0 <
1, u0 ( ) > 0, and u00 ( )
P1
Xt . Only outlays Xt induce tax distortions.
t
t=0
(3)
u (Ct )],
0. Isolating private consumption and using the
government budget constraint to substitute out debt and taxes yields:
Ct = Yt + q (Wt
Xt )
h (Tt )
(4)
Gt :
The government’s problem is to choose q and the path of taxes (equivalently the path of
government debt) to maximize household utility
U = E0 [
P1
t=0
t
u (Yt + q (Wt
Xt )
h (Tt )
Gt )],
(5)
taking the path of fYt ; Wt ; Xt ; Gt g as given. In other words, the government faces a stochastic
endowment fYt g and production technology fWt ; Xt g as well as a set of stochastic government
expenditures fGt g :
2.1.1
Examples
To make the setup concrete, we brie‡y discuss two examples. Appendix A provides a more
explicit formal mapping of these examples into our model setup. The …rst example comes from
…nancial economics and considers the value of government interventions to prevent economically
destabilizing …re sales, using Stein’s (2012) model as a baseline. In that model, runs by shortterm creditors can result in …re sales. The real cost of these …re sales is that outside investors
must use capital to purchase liquidated assets, rather than investing in productive new projects.
In this setting, consider a program where the government steps in and guarantees the shortterm debt in order to prevent runs. For simplicity, assume these guarantees ensure there is
in Yt . For instance, in the case of deposit insurance, Wt represents incremental social bene…ts of deposit
insurance, taking as given the existence of a lender of last resort.
5
no run, and further assume the guarantee is unanticipated so that it has no ex-ante e¤ects.
In our notational conventions, the government’s outlays X are the realized …scal costs of the
guarantee, which will be positive if the government has to pay to make the short-term debt
whole. The social payo¤ W from this program is the gain in net private income created by the
fact that outside investors can use their capital for productive new projects.
Our second example considers unemployment insurance, taking the Rothschild-Stiglitz (1976)
model as our baseline. In that model, there are two types of agents with di¤erent probabilities
of becoming unemployed.2 It is well-known that when agents know their types, pooling competitive equilibria may not exist. In the separating equilibrium that can exist, agents with a low
probability of future unemployment must be less than fully insured to prevent agents with a
high probability of future unemployment from mimicking them and purchasing cheap insurance.
In this setting, government-run mandatory insurance can improve overall welfare by enforcing
pooling. The social payo¤ W from mandatory insurance is the welfare gain associated with the
move from the separating equilibrium to the pooling outcome. And the net government outlay
X is the di¤erence between government insurance payouts and premia collected.
2.1.2
Discussion of Setup
Several features of the model deserve discussion. First, note that since we are working in
a representative agent framework, all income here is “social” in the sense that it ‡ows to the
representative agent. Social bene…ts Wt ultimately ‡ow to the representative agent even though
they may not be taken into account when individual agents make decisions. Note that if market
failures create scope for the government policies to a¤ect Pareto improvements, a representative
agent may fail to exist. In the case of simple incomplete markets problems like pollution
externalities, a representative agent may exist. However, in other cases where market failures
are generated by information problems among heterogeneous agents, it may not exist (Huang
and Litzenberger, 1988; Du¢ e, 2001). In cases where no formal representative agent exists, our
framework can be viewed as a short-hand for a planner maximizing a more complicated social
welfare function. Our key simplifying assumption then, which may not always be satis…ed, is
that aggregate consumption is a su¢ cient statistic for social welfare. The intuitions that emerge
from our model should still apply in more complex settings where such simple aggregation fails.
Programs that generate social bene…ts when all agents have low consumption will be valuable
social hedges. Similarly, programs that increase the government tax burden at such times will be
…scally risky. However, our framework will be unable to capture intuitions where distributional
2
As we discuss further below, there is some tension in our use of a representative agent framework in situations
where there are multiple types of agents.
6
considerations are important. For instance, in the case of deposit insurance, there is no sense
in which banks and their shareholders, rather than consumers, are paying insurance premia in
our setup.
A second feature of the model setup is that distortions apply to the level of taxes as opposed
to the tax rate. Formally, this means that taxes are distortionary despite the fact that they are
modeled as lump-sum in our setup. We adopt this speci…cation solely for analytical tractability.
This is a simple modeling device that allows us to abstract from the types of complex optimal
taxation problems studied in the public …nance literature (Stokey and Lucas, 1983; Aiyagari,
Rao, Sargent, and Seppala, 2002; Golosov, Tsyvinski, and Werning, 2006; Acemoglu, Golosov,
and Tsyvinski, 2008, among many others), which are not our focus, and enables us to obtain
closed-form solutions. And as we show in Section 2.4 below, the exact same basic forces obtain
in a model in which distortions are a function of tax rates. Furthermore, although we refer to
distortions h (T ) as “tax distortions,” they serve as a short-hand for a host of costs that are
incurred when taxation is large or when the government faces a signi…cant …scal burden (e.g.,
frictions associated with the risk of sovereign default).
A third feature of the set up is that the government makes a one-shot choice about program
scale and can credibly commit to this scale inde…nitely. Thus, the model is best seen as applying
to non-discretionary budgeting where, for reasons of e¢ ciency, fairness, or political economy,
program scale is stable over time. In practice, we will assume that these decisions are made
from the vantage point of the economy’s steady-state. However, in Section 5 we brie‡y explore
the other extreme of a pure discretionary policy where the government is free to adjust the
desired scale each period.
Finally, the setup largely abstracts from the notion that the presence of a government
project can change private behavior. Formally, by taking Yt as …xed, we are assuming that the
private sector does not react to the government’s choice of project scale q. This is potentially
important since many critiques of government programs, particularly guarantee programs, argue
that they distort private sector behavior. For instance, deposit insurance may create a moral
hazard problem, increasing bank risk taking beyond the social optimum. In the context of the
model, one could think of these kinds of e¤ects as being captured in the W s and Xs, but they
are not explicitly modeled here.
7
2.2
The Key First Order Condition
We now solve the model, deriving the …rst order condition for the optimal scale of the project
q. The …rst order condition for optimal issuance of t-period bonds at time 0 is
(t)
0 = u0 (C0 ) h0 (T0 )
for all t
E0 [ u0 (Ct ) h0 (Tt ) R0 ]
(6)
1. To see this, …x a time path of government borrowing and taxation. Consider a
(t)
deviation in which the government issues more riskless t-year debt D0 and reduces taxes Tt
by the same small amount. This deviation reduces tax distortions by h0 (T0 ) at time 0, which
raises utility at time 0 by u0 (C0 ) h0 (T0 ) at the margin. Since this deviation raises future taxes
(t)
(t)
by (R0 )t at time t, it raises future tax distortions by h0 (Tt ) (R0 )t at time t, which lowers
(t)
discounted expected utility by E0 [ u0 (Ct ) h0 (Tt ) (R0 )t ] at the margin. Equation (6) says that,
along the optimal path of debt and taxes, such a deviation must have zero e¤ect on discounted
expected utility.
Letting
M0;t =
t 0
u (Ct )
u0 (C0 )
(7)
(t)
denote the stochastic discount factor for t-period ahead payo¤s at time 0 and recalling that R0 ,
(t)
the gross yield on t-period riskless borrowing from time 0 to t, satis…es (R0 )t = 1=E0 [M0;t ], we
can rewrite the …rst order condition for borrowing as
h0 (T0 ) =
for all t
E0 [M0;t h0 (Tt )]
E0 [M0;t ]
(8)
1.
We next solve for the optimal scale q of the government project. Note that the e¤ect of
changing q on household consumption at time t is given by
@Ct
= Wt
@q
Xt (1 + h0 (Tt )) .
Increasing project scale q directly alters the contribution of the government project to consumption by (Wt
t
Xt ) and increases tax distortions by h0 (Tt ) @T
. The main analytical bene…t from
@q
specifying tax distortions as a function of the level of taxes is that
@Tt
@q
takes a simple form: it
is simply equal to Xt . Overall, the net impact of increasing the scale of the government project
is Wt
Xt (1 + h0 (Tt )) :
8
The optimal amount of government activity, q , satis…es
P1
0=
which we can rewrite as
0=
P1
t=0
t
t=0
E0 [u0 (Ct )
@Ct
],
@q
Xt (1 + h0 (Tt )))]:
E0 [M0;t (Wt
(9)
Proposition 1 The optimal scale of government intervention q is implicitly de…ned by
Expected cash‡ow cost/bene…t
z
}|
P1 E0 [Wt
0 =
t=0
P1
(t)
{
Xt ]
(R0 )t
E0 [h0 (Tt ) Xt ]
t=0
|
(t)
(R0 )t
{z
+
Cash ‡ow risk premium
}|
Cov
[M
0
0;t ; Wt
t=0
zP
1
{
Xt ]
(10)
P1
Cov0 [M0;t ; h0 (Tt ) Xt ],
| t=0
{z
}
}
Tax risk premium
Expected tax cost/bene…t
(t)
(t)
where we have used the fact that E0 [M0;t ] = 1=(R0 )t where R0 is gross yield on t-period riskless
bonds from time 0 to t.
(t)
Proof. Expand (9) and use the identity E0 [M0;t Zt ] = (R0 ) t E0 [Zt ] + Cov0 [M0;t ; Zt ] :
We can use (10) to pin down the optimal scale of some government program, taking its
(possibly stochastic) …scal costs Xt and social bene…ts Wt as given. However, as discussed
below, we can also use (10) to pin down the optimal “pricing”of a government program, taking
a desired program scale as given.3
2.2.1
Interpretation
First, we note that if Wt = 0 so there is no social payo¤ from the government project and
h0 (Tt ) = 0 so there are no tax distortions, then expression (10) collapses to the standard asset
pricing equation
0=
P1
t=0
E0 [M0;t Xt ] =
P1
(t) t
t=0 (R0 ) E0
[Xt ] +
P1
t=0
Cov0 [M0;t ; Xt ] .
(11)
Under these assumptions, our model collapses to the frictionless benchmark where the government evaluates projects just as the private sector would. Relative to the frictionless benchmark,
our model contemplates the possibility that government projects may deliver social bene…ts
3
This is similar to asset pricing where the same Euler equation can be used (i) to pin down optimal asset
holdings taking returns as given or (ii) to pin down equilibrium returns taking asset supply as given.
9
by mitigating market failures (Wt > 0), but do so in the presence of distortionary taxation
(h0 (Tt ) > 0).
There are four terms in expression (10) for the optimal scale of a government program which
we now discuss in turn.
Expected cash‡ow cost/bene…t The …rst term in expression (10) is the expected net
cash ‡ow bene…t from the government program, discounted at the risk-free rate. This term
encapsulates the way the cost of credit and guarantee programs is accounted for in the Federal
budget. Speci…cally, under the Federal Credit Reform Act (FCRA) of 1990,4 the expected net
present value of government outlays discounted at the risk-free rate for such programs should
be zero. In the notation of the model, the expected NPV of Xt for guarantee programs should
be zero. The model shows that the total e¤ect, including the social gain from the program
(Wt ) and the direct cash ‡ows associated with the program (Xt ), should be taken into account
from a cost-bene…t perspective. In particular, the e¤ect of the program on the unconditional
path of private output could be positive if the program …xes some unconditional market failure
(E0 [Wt ] > 0).
Cash ‡ow risk premium The second term in (10) is the risk premium associated with
these cash ‡ows. It is commonly argued (e.g., Lucas (2012) and the citations within) that the
government should charge the same risk premium as the private sector because it is acting
on behalf of risk-averse tax payers. Speci…cally, for bearing the risk of cash outlays Xt , the
government should charge the risk premium Cov0 [M0;t; Xt ] just as private investors would.
Again, our model shows that the risk premium should be associated with the total social gain
or loss from the project. Speci…cally, the term Cov0 [M0;t ; Wt ] captures the government’s “social
risk management”motive. In particular, for some government programs like deposit insurance,
the social gains from correcting market failures may accrue primarily in bad times. Despite
the fact that the government takes cash ‡ow losses from deposit insurance in bad times, the
program also generates broad social and economic gains at exactly those times. The …rst e¤ect
makes deposit insurance risky for the government, but the second e¤ect makes it desirable
to o¤er. Essentially, deposit insurance is a desirable social hedge, Cov0 [M0;t ; Wt ] > 0, which
reduces the risk premium that the government should charge for it.
One subtlety regarding this term, is that the stochastic discount factor M0;t and therefore
risk premia depend on the scale of the government program q and taxes. This can be seen
4
See http://www.fas.org/sgp/crs/misc/R42632.pdf
10
clearly if we substitute (4) into (7) to obtain
M0;t =
t 0
u (Yt + q (Wt Xt ) h (Tt ) Gt )
.
(Y0 + q (W0 X0 ) h (T0 ) G0 )
u0
Intuitively, the consumption of the representative household is a¤ected by the existence of
government programs and tax distortions. If government policies reduce (increase) the volatility
of consumption and, hence, the volatility of marginal utility, risk premia will be lower (higher)
than they would in the corresponding economy where q = 0.
Expected tax cost/bene…t The third term in (10) captures the government’s “…scal risk
management”motive. When h0 (Tt ) > 0, tax distortions induce the government to act as if it is
more risk-averse than the taxpayers it represents. Intuitively, because project outlays Xt ‡ow
through the government budget, they a¤ect taxes and therefore tax distortions. Speci…cally,
since E0 [h0 (Tt ) Xt ] = E0 [h0 (Tt )] E0 [Xt ] + Cov0 [h0 (Tt ) ; Xt ], this term makes programs less
desirable to the extent they tend to raise taxes on average (E0 [Xt ] is large) or tend to raise
taxes at times when taxes would already be elevated (Cov0 [h0 (Tt ) ; Xt ] is large). By contrast,
if h0 (Tt ) = 0, the model collapses to the Ricardian case in which the government is a veil for
taxpayers.
One way to understand the government’s …scal risk management motive is to note that
the discount factor for the additional private income generated by a government program is
W
= M0;t ; while the discount factor for the government outlays/inlays associated with the
M0;t
X
= M0;t (1 + h0 (Tt )). Since M0;t > 0 and h0 (T ) > 0 for all T > 0, it follows that
program is M0;t
W
X
we have M0;t
< M0:t
. This means that, all else equal, the government should apply a lower
discount rate to government outlays Xt than to social bene…ts Wt . The fact that MtW < MtX
re‡ects the …scal conservatism that is built into our framework due to the distortionary costs
of taxation.
As a second illustration of this motive, consider a traditional …scal stimulus program. Assume that this program requires government spending of Xt > 0 and generates additional
private output of Wt = ft Xt > 0 where ft is the “…scal multiplier.” In a Ricardian world, the
government should undertake the program if Wt
Xt > 0, which is equivalent to ft > 1. How-
ever, with distortionary taxation, it should only undertake the program if Wt > (1 + h0 (Tt )) Xt ,
which corresponds to ft > 1 + h0 (Tt ) > 1: Intuitively, …scal multipliers must be greater than
one in order to compensate for the distortionary costs of taxation.
A critical tension that emerges from our framework is the con‡ict between the social risk
management motive captured by the second term in (10) and the …scal risk management motive
11
captured by the third term. Programs like deposit insurance and automatic stabilizers that have
signi…cant social risk management bene…ts tend to involve government expenditures and, hence,
higher tax distortions in bad times, creating more …scal risk.
Tax risk premium The …nal term in (10) is the risk premium associated with the cyclicality
of tax distortions. If a program leads to increased tax distortions in bad economic times, these
distortions reduce private consumption precisely when it is most valuable. Thus, the term
Cov0 [M0;t ; h0 (Tt ) Xt ] re‡ects the interaction between the “social risk management”and “…scal
risk management”motives.
A Taxonomy of Government Programs The four forces captured in our key …rst order condition (10) provide a taxonomy of government programs along two dimensions. The
…rst is whether the associated government outlays carry aggregate risk. In the terminology
of our model, programs with systematically risky outlays have Cov [Mt ; Xt ] 6= 0. This type
of risk is closely related to though not exactly the same as …scal risk, which is systematic if
Cov [Mt ; h0 (Tt ) Xt ] 6= 0 in the terminology of the model. The second dimension along which
programs di¤er is whether the social payo¤s they generate for society vary systematically. In
our terminology, programs with systematic social cash ‡ows have Cov [Mt ; Wt ] 6= 0. It is worth
noting at this point that the social bene…ts of government programs Wt can be di¢ cult to
measure, and their covariances with marginal utility may be even more di¢ cult to measure in
practice. Nonetheless, it is straightforward to qualitatively classify programs according to the
cyclicality of their required outlays and social bene…ts.
At one end of the spectrum are programs that have idiosyncratic government outlays and
idiosyncratic social cash ‡ows. Programs in this category include government ‡ood insurance,
government terrorism insurance, and government agricultural insurance. The incidences of
‡oods, terrorist attacks, and crop failures are largely uncorrelated with the behavior of the
macroeconomy as a whole. It is sometimes argued that government involvement in these areas
is necessary because private insurers become decapitalized after a disaster, making it di¢ cult
for consumers and …rms to purchase new insurance after the disaster. However, such bene…ts are not strongly correlated with the state of the economy and are therefore idiosyncratic.
Though these programs may provide idiosyncratic social bene…ts, neither their impact on the
government budget nor their social bene…ts vary systematically with aggregate consumption.
For these types of programs, our model shows that a modi…ed version of the Arrow-Lind
(1970) theorem holds. As in Arrow-Lind, the government should use the risk-free rate as its cost
of capital. However, in contrast to Arrow-Lind, our model shows that this discount rate should
12
be applied to the social net cash ‡ows generated by the program, including the uninternalized
social cash ‡ows and the marginal tax distortions it generates. In the notation of the model,
the relevant cash ‡ows are given by (Wt
Xt (1 + h0 (Tt ))) :
At the other end of the spectrum are programs with systematic government outlays and
systematic social bene…ts. Programs in this category include …nancial stabilizers like deposit
insurance and lender of last resort activities, as well as automatic stabilizers for the real economy
including unemployment insurance. Such programs require government outlays in bad times
but may also deliver signi…cant social bene…ts at those times. For instance, as documented by
Calomiris and Gorton (1990) among others, the risk of bank runs is highest during economic
downturns. Deposit insurance and the lender of last resort activities prevent the ine¢ cient
early liquidation of long-term projects (e.g., Diamond and Dybvig, 1983) at such times, raising
overall social output in bad states of the world. Thus, these activities can provide strong macroeconomic hedging bene…ts. However, these social hedging bene…ts come at a cost. Government
expenditures on these activities are strongly countercyclical, which means that the government
is incurring direct costs as well as the costs of marginal tax distortions in bad states of the
world.
For these types of programs, the Arrow-Lind (1970) theorem does not hold, even in a
modi…ed form. The government must charge a risk premium. This risk premium may be either
positive or negative. It is negative if the social hedging bene…ts of the program outweigh the
riskiness of the direct costs and tax distortions. Otherwise it is positive. Again, in the notation
of the model, the relevant cash ‡ows are (Wt
Xt (1 + h0 (Tt ))) :
Finally, there are intermediate programs that require systematically risky government outlays but generate idiosyncratic social cash ‡ows. Take for example the case of student loan
guarantees. The standard rationale for these guarantees is that they provide unconditional
social bene…ts; since human capital cannot easily be pledged, the private supply of student
loans may be lower than the …rst best level. The bene…ts that government guarantees provide
in terms of expanding the supply of student loans accrue continually and do not have a strong
correlation with the business cycle. Thus, the social bene…ts of guarantees are idiosyncratic.
However, since student loan defaults tend to happen in recessions, the private cash ‡ows from
the program are risky.
2.3
Optimal Program Scale
In the case where h (T ) = ( =2) T 2 , we can derive closed-form expressions for taxes, debt, and
the optimal scale of government involvement q. We begin by outlining the standard solution for
optimal taxation (equivalently optimal debt management) in an in…nite horizon setting when
13
taxes are distortionary.
Let Et [ ] denotes expectations under the risk-neutral measure. Speci…cally, for any random
(t)
variable Z we de…ne E0 [Zt ] = E0 [(M0;t =E0 [M0;t ]) Zt ] = E0 [M0;t Zt ] R0 , which implies that
(t)
E0 [M0;t Zt ] = (R0 ) t E0 [Zt ] : Straightforward arguments show that the government’s lifetime
budget constraint at time t is
P1
(j) j P1
(i) i (i)
j=0 (Rt ) [
i=j+1 (Rt i ) Dt i ]
=
P1
(j) j
j=0 (Rt ) Et
[Tt+j
(Gt+j + qXt+j )] :
(12)
Equation (12) simply says that the market value of outstanding government debt (the left
hand side) equals the discounted risk-adjusted present value of future primary surpluses. If
h (T ) = ( =2) T 2 , then (8) implies that
(13)
Tt = Et [Tt+j ]
for all j
1. In other words, tax-smoothing implies that tax distortions and, hence, taxes
are a random walk under the risk-neutral measure. Combining (12) and (13), we see that the
optimal level of taxes is
Tt =
P1
(j) j
(i) i (i)
(j) j P1
j=0 (Rt ) Et
i=j+1 (Rt i ) Dt i ] +
j=0 (Rt ) [
P1
(j) j
j=0 (Rt )
P1
[Gt+j + qXt+j ]
:
(14)
Since the denominator of (14) is the price of a perpetuity that pays one unit each period,
equation (14) states that taxes should equal the annuitized value of the accumulated debt
burden plus the discounted risk-adjusted present value of future expenditures.
With this solution for optimal taxes in hand, we can express the optimal scale of the government q in closed form. Speci…cally, using (13), we can rewrite the …rst order condition (9)
as
0=
P1
(t) t
t=0 (R0 ) E0
[Wt
Xt ]
P1
(t) t
t=0 (R0 ) E0
14
[Xt ] T0
P1
(t) t
t=0 (R0 ) Cov0
[Xt ; Tt ] : (15)
Substituting (14) into (15) and solving, we obtain an expression for the optimal level of q :
q
=
8
>
>
>
>
>
<
>
>
>
>
>
:
P1
(t) t
Xt ]
0 [Wt
t=0 (R0 ) hE
P
P
i P1
1
1
(t)
(t) t (t)
(t)
t
P1
(R
)
(R
)
D
+
(R0 )
0
0
t
0
t
(t) t
t=0
i=t+1
t=0
P
[X
]
(R
)
E
1
t
(t)
0
0
t=0
(R0 ) t
t=0
P1 (j) hP1
i P1
(i) i (i)
(j)
j
P1
(Rt )
(Rt i ) Dt i +
(Rt )
(t) t
j=0
i=j+1
j=0
P
X
;
(R
)
Cov
1
t
(j)
0
0
t=0
8
< P
(t)
1
t=0 (R0 )
:
t
P1
(t) t
t=0 (R0 ) E0 [Xt ]
P1
(t) t
t=0 (R0 )
!2
j=0
+
(Rt )
t E [G ]
t
0
jE
t
[Gt+j ]
j
P1
(t) t
t=0 (R0 ) Cov0
"
Xt ;
P1
9
>
>
>
>
>
=
>
>
>
>
>
;
(j) j
j=0 (Rt ) Et [Xt+j ]
P1
(j) j
j=0 (Rt )
Equation (16) is the ratio of the two terms in curly braces, and it bears a close resemblance
to standard portfolio choice expressions in asset pricing (see e.g., Campbell and Viceira, 2003).5
The denominator of (16) depends on the sum of the square of the lifetime average risk-adjusted
outlays plus the covariance of outlays with future average lifetime outlays, all times the strength
of distortions . The denominator is the analog of the return variance times risk aversion
in standard asset pricing models. Here, marginal tax distortions
induce a form of …scal
risk aversion which pins down the optimal scale of government programs. (This is distinct
from the household’s risk aversion which is embedded in the risk neutral expectations E0 [ ])
Loosely speaking, since tax distortions are convex and E0 [Xt2 ] = (E0 [Xt ])2 + V ar0 [Xt ], all else
equal, the government dislikes projects with large average expenditures or with highly volatile
expenditures.
The numerator of (16) contains three terms. First,
P1
(t) t
t=0 (R0 ) E0
[Wt
Xt ] is the expected
present value of the net social payo¤ to the project— the social payo¤ minus the associated
government outlays— which is the analog of the expected excess return in standard portfolio
choice settings. The next two terms capture the government’s …scal risk management motive.
The …rst of these captures the fact that, while the government always dislikes projects with
high levels of expected lifetime outlays, this dislike becomes more pronounced when (i) the
accumulated debt burden is high or (ii) when exogenously given lifetime government outlays
are high. The second of these captures the fact that the government dislikes programs that add
to the tax burden in states when the burden would otherwise already be high. These terms are
analogous to those that appear in portfolio choice settings in which investors are endowed with
“background risk”(e.g., labor income risk).
5
Since the SDF depends on q, this is not a closed-form since both the risk-neutral measure as well as
riskless interest rates are potentially an implicit function of q. Furthermore, future realizations of the value of
outstanding debt will also depend implicitly on q.
15
(16)
#9
=
;
:
2.4
Distortions as a Function of the Tax Rate
Before studying the above model in greater detail, we …rst con…rm that specifying distortions
as a function of the level of taxes as opposed to the tax rate has little e¤ect on our basic
conclusions. A vast literature in public economics shows that income taxation and other forms
of proportional taxation give rise to excess burden because they inevitably distort behavior in
ine¢ cient ways. By contrast, lump sump taxation creates no such excess burden in conventional
models. Thus, we want to be sure that this modelling convenience made for the sake of analytical
tractability has little e¤ect on our overall conclusion.6 Here we show that a very similar set of
trade-o¤s emerges if we instead model distortions as a function of the tax rate.
The government’s budget constraint is as above in (1). However, we now assume that
government levels a proportional tax
t
on total private income which raises tax revenues of
Tt =
t
(Yt + qWt ) .
Critically, we now assume that the household budget constraint is
(Yt + qWt ) (1
h ( t )) +
P1
(j) j (j)
j=1 (Rt j ) Dt j
= Ct + Tt +
P1
j=1
(j)
Dt ,
where h ( ) represents the distortionary costs or excess burden of taxation with h0 ( ) > 0 and
h00 ( ) > 0. This speci…cation for the excess burden of taxation follows Barro (1979) and Bohn
(1990). As above, a natural choice here is h ( ) = ( =2)
In this setting, the …rst order condition for
h0 ( 0 ) =
for t
(t)
D0
2
.
implies that optimal taxation satis…es
E0 [M0;t h0 ( t )]
E0 [M0;t ]
1. As above, tax-smoothing implies that distortions are a random walk under the
risk-neutral measure. Similarly, the optimal amount of government activity satis…es
0 = E0 [M0;t Wt (1 + h0 ( t )
t
h ( t ))]
6
E0 [M0;t Xt (1 + h0 ( t ))]
Why does the h (T ) = ( =2) T 2 assumption buy us signi…cant analytical tractability relative to a h ( ) =
( =2) 2 assumption? In short, the h (T ) = ( =2) T 2 assumption makes it possible to obtain closed-form solutions
for the optimal path of taxes, which can then be used to obtain closed-forms for q . By contrast, with the
h ( ) = ( =2) 2 assumption it is very di¢ cult to obtain closed-form solutions for tax rates (Bohn, 1990).
16
or
Expected cash‡ow cost/bene…t
risk premium
}|
{ z Cash ‡ow }|
{
P
(t) t
1
0 =
(R
)
E
[W
X
]
+
Cov
[M
;
W
X
0
t
t
0
0;t
t
t]
0
t=0
t=0
P
(t) t
0
+ 1
h ( t )) Xt h0 ( t )]
t=0 (R0 ) E0 [Wt (h ( t ) t
|
{z
}
z
P1
(17)
Expected tax cost/bene…t
P
0
+ 1
t=0 Cov0 [M0;t ; Wt (h ( t )
|
{z
t
h ( t ))
Tax risk premium
Xt h0 ( t )] .
}
Comparing (17) with (10) from above, we see that the same basic set of forces emerges where
we assume that distortions are tied to the level of taxes or tax rates.
However, there are some minor di¤erences that are worth noting. Speci…cally, the disW
=
count factor the government applies to the private income generated by a program is M0;t
M0;t (1 + h0 ( t )
t
h ( t )), and the discount factor applied to the government inlays/outlays
X
= M0;t (1 + h0 ( t )). Since 0 < h0 ( t )
associated with the program is M0;t
for all
t
t
t
h ( t ) < h0 ( t )
W
X
2 (0; 1] by the convexity of h ( ), it follows that we have M0;t < M0;t
< M0;t
for all
2 (0; 1]. This means that the government should apply a lower discount rate to Xt than to
W
X
re‡ects the …scal conservatism built into our frameWt . As above, the fact that M0;t
< M0;t
work due to the distortionary costs of taxation.7,8 However, when tax distortions depend the
tax rate, the government should also apply a lower discount rate to the social bene…ts of the
W
program than private agents would (i.e., M0;t < M0;t
).
What is the intuition for this result? And how does this compare to the solution in the
X
W
X
W
< M0;t
= Mt (1 + h0 (Tt )), so that M0;t
= M0;t and M0;t
baseline model? Above we had M0;t
for Tt > 0. Thus, relative to the case where distortions are linked to the level of taxes, when
distortions are a function of the tax rate, the government places additional weight on private
income generated by government programs, Wt . Intuitively, there is an additional incentive to
manage social risk: this keeps the tax base high, which keeps tax rates low, which reduces tax
7
Obviously, @M0;t =@ t > 0 because tax distortions reduce consumption, private agents place greater weight
X
W
=M0;t =@ t > 0 and @ M0;t
=M0;t =@ t > 0;so a
on states with high tax rates. However, we have @ M0;t
benevolent government evaluating the impact of a program on private output and government outlays places
even more weight on high tax states.
8
As above, consider a traditional …scal stimulus program that requires goverment spending of Xt > 0 and
generates private output of Wt = ft Xt > 0. In a Ricardian world, the government should undertake the
project if ft > 1. However, with distortionary taxation, the government should only undertake the project if
ft > [1 + h0 ( t )] = [1 + h0 ( t ) t h ( t )] > 1: The numerator (1 + h0 ( t )) re‡ects that fact that government
outlays are costlier than they would be in a Ricardian world since they must be …nanced with distortionary
taxes. The denominator (1 + h0 ( t ) t h ( t )) re‡ects the fact that, when distortions are linked to the tax rate
as opposed to level, the government wants to expand the tax base because this allows it to keep the tax rate
and thus distortions down. However, the net e¤ect here is always towards …scal conservatism.
17
distortions. In e¤ect, the desire to manage …scal risk ampli…es the desire to manage social risk.9
With this single caveat in mind, we conclude that little of substance is changed by our
modeling convenience of linking distortions to the level of taxes as opposed to tax rates. And,
as we shall see in the next section, the payo¤ is that we can obtain simple closed-forms that
provide deeper intuition into the key trade-o¤s we are after.
3
Special Cases of the General Model
To simplify the analysis and build intuition, we consider two special cases of the general model
where the assumption that h (T ) = ( =2) T 2 in combination with other assumptions enables us
to obtain a full closed-form solution. These solutions in turn allow us to easily characterize the
key comparative statics of the model. The …rst special case is a two-period version of the model
allowing for general household risk aversion. The second special case is an in…nite horizon
version of the model, imposing risk neutrality and AR(1) speci…cations for the evolution of all
exogenous state variables.
3.1
3.1.1
Two-Period Model with Risk Aversion
Set Up and Solution
The …rst special case we consider is a simple two-period version of the model. In this case, the
government chooses program scale q as well as the amount of 1-period debt to be issued D0 at
time 0; all debt must be repaid at time 1. For simplicity, we assume h (T ) = ( =2) T 2 . We also
assume, without loss of generality, that the government begins time 0 with no debt.
In the two-period model, the condition for optimal time 0 borrowing is T0 = E0 [T1 ]. Using
the accounting de…nitions of taxes T0 =
D0 =
where R = (E0 [M0;1 ])
1
G0
D0 + G0 + qX0 and T1 = RD0 + G1 + qX1 , this implies
E0 [G1 ] + q (X0
1+R
E0 [X1 ])
,
is the gross interest rate between time 0 and 1. In other words, the
government chooses D0 to smooth risk-adjusted expenditures Gt and Xt across periods. The
solution for optimal borrowing can be substituted into the accounting de…nitions of taxes to
9
Note, however, that this additional weight on Wt only applies to government programs that alter taxable
private income. Examples here might include macro-prudential policies aimed at controlling …nancial crises or
…scal stimulus programs or automatic stabilizers that have a multiplier e¤ect.
18
obtain
G0 + R 1 E0 [G1 ] + q (X0 + R 1 E0 [X1 ])
1+R 1
G0 + R 1 G1 + q (X0 + R 1 X1 ) + (G1 E0 [G1 ]) + q (X1
=
1+R 1
T0 =
T1
E0 [X1 ])
,
which clearly satisfy the optimality condition for debt and taxes since T0 = E0 [T1 ]. Since there
only two periods, all time 1 uncertainty is borne by taxes at time 1.
In the two-period case, the key Euler equation (9), which determines the optimal scale of
government involvement in the project, reduces to
0 = [W0
X0 (1 + T0 )] + R 1 E0 [W1
X1 (1 + T1 )] :
(18)
Finally, substituting the above expressions for taxes into the optimality condition for q and
solving implies
q =
(W0
X0 ) + R 1 E0 [W1
X1 ]
(G0 +R
(X0 +R 1 E0 [X1 ])2
1+R 1
1 E [G ])(X +R 1 E [X ])
1
0
1
0
0
1+R 1
R 1 Cov0 [X1 ; G1 ]
.
+ R 1 V ar0 [X1 ] .
(19)
Equation (19) is the precise 2-period analog of the more complicated solution for the in…nite
period model given in equation (16).
3.1.2
Comparative Statics
We now derive comparative statics in the two-period model. Here we state the comparative
statics, taking M0;1 as given so that we do not take into account how an endogenous change
in the scale of the government program impacts risk premia via changes in the SDF. This is
equivalent to assuming that the project is small relative to private consumption, which is a
reasonable assumption for some, but not all, government programs. In particular, if we wanted
to capture the idea that the …nancial stability bene…ts of deposit insurance lower risk premia
in general, or the Pastor and Veronesi (2013) idea that uncertainty about government actions
creates its own risk premium, we could not take M1 as given.
Formally, our comparative statics holding the SDF …xed but allowing net household consumption to change are analogous to substitution e¤ects in the context of a standard demand
function. Conversely, the e¤ects holding net household consumption constant but shifting the
SDF are analogous to income e¤ects. Since our setting is analogous to traditional consumptionsaving decision, income e¤ects will generally have the opposite sign of substitution e¤ects. As
19
is always the case, without restrictions on preferences, either income or substitution e¤ects can
dominate. Of course, we usually think that substitution e¤ects dominate in consumption-saving
decisions (e.g., people save more when expected returns rise), and this will be true under various parametric restrictions. Loosely speaking, this is always true in our model so long as risk
aversion is not too high. This is clear since in the limiting case of risk neutrality, the SDF is
constant and thus independent of q, so there are no income e¤ects. The Internet Appendix
provides a more detailed analysis of these issues.
The following proposition states the e¤ects of changing various parameters on the optimal
scale of the project q .
Proposition 2 In the two-period model, assume that h (T ) =
1
2
T 2 and G0 + R 1 E0 [G1 ]
0.
Then we have the following comparative statics (holding …xed the SDF):
@q =@W0 > 0 and @q =@E0 [W1 ] > 0,
@q =@X0 < 0 and @q =@E0 [X1 ] < 0;
@q =@G0 /
@q =@ /
(X0 + R 1 E0 [X1 ])=(1 + R 1 ),
T0 ((X0 )+R 1 E0 [X1 ]) R 1 Cov0 [X1 ; T1 ] =
1
((W0
X0 ) + R 1 E0 [W1
@q =@Corr0 [X1 ; G1 ] < 0.
In general, the sign of @q =@V ar0 [X1 ] depends on sign (q ) and Corr0 [X1 ; G1 ]. However,
in the common case where q > 0 and Corr0 [X1 ; G1 ] > 0, we have @q =@V ar0 [X1 ] < 0.
Proof. Di¤erentiation of equation (19).
The …rst two comparative statics are intuitive. Raising the social payo¤ to the project at
time 0, W0 , or the risk-adjusted expected externality return at time 1, E0 [W1 ], increases the
optimal scale of the project. Since R 1 E0 [W1 ] = R 1 E0 [W1 ] + Cov [M1 ; W1 ] ; the discounted
risk-adjusted expected externality return at time 1 can be raised either by increasing expected
social payo¤ E0 [W1 ] under the physical measure or by making the project a better social hedge
by increasing Cov0 [M1 ; W1 ]. Similarly, raising the required government outlays for the projects
at time 0, X0 , or the risk-adjusted outlays at time 1, E0 [X1 ], lowers the optimal scale of the
project. E0 [X1 ] can be raised either by increasing expected outlays E0 [X1 ] under the physical
measure or by having outlays that occur at bad economic times (i.e., raising Cov0 [M1 ; X1 ].).
When
> 0, the e¤ect of initial government spending G0 depends on whether the program
is expected to generate net government outlays or inlays in risk adjusted terms. In particular,
if we have X0 + R 1 E0 [X1 ] > 0, then the program is expected to entail net outlays for the
20
X1 ])
government in risk-adjusted terms. The optimal scale of positive-outlay programs declines
with G0 . Intuitively, increasing G0 increases the government’s …scal burden and therefore tax
distortions. By decreasing the scale of a positive outlay program, the government can reduce
the need for taxation partially o¤setting the e¤ect of increasing G0 . In contrast, if we have
X0 + R 1 E0 [X1 ] < 0 so that programs entail risk-adjusted expected inlays, then the optimal
scale of the project rises as G0 rises. As the government’s …scal burden rises, we want to increase
the scale of activities that reduce that burden.
Similar logic applies to the e¤ect of the severity of marginal tax distortions, . The comparative static has two terms. The …rst,
T0 ((X0 ) + R 1 E0 [X1 ]), depends on the risk-adjusted
expected cash outlays of the program and time 0 taxes. If the program requires positive outlays, then decreasing the scale of the program reduces the tax burden. Reducing the tax
burden is particularly valuable when the level of taxes, T0 , is already high because marginal
tax distortions are large at high levels of taxation. The second term in the comparative static,
R 1 Cov0 [X1 ; T1 ], indicates that the optimal scale of the program also decreases if it tends to
typically require outlays at times when the time 1 tax burden will be high. As tax distortions
increase, programs that covary with other …scal risks become less attractive to the government.
At the optimum, @q =@ is also proportional to
increase in distortions
1
((W0
X0 ) + R 1 E0 [W1
X1 ]), so an
leads the government to cut back on attractive projects with large
discounted, risk-adjusted net bene…ts.
Finally, the government does less of programs whose t = 1 outlays are (i) highly correlated
with G1 and (ii) more variable (under regularity conditions).
We next consider comparative statics on the price that the government should be willing
to pay for di¤erent projects. Speci…cally, we can …x the supply of the project at q and then
allow the initial price of the project P = X0 to adjust so that demand q equals supply q. The
notational convention here is that P is the initial price the government pays to undertake a
project. Speci…cally, using (18), we have
P0 =
W0 + R 1 E0 [W1 X1 (1 + T1 )]
.
1 + T0
The following proposition characterizes the behavior of government program prices.
Proposition 3 In the two-period model, assume that W0 = 0 and h (T ) =
the following comparative statics (holding …xed the SDF):
@P =@W0 > 0 and @P =@ (E0 [W1 ]) > 0
@P
@G0
/
( =(1 + R 1 ))(P + R 1 E0 [X1 ]),
21
1
2
T 2 : Then we have
@P
@
/
@P
@q
/
T0 (P + R 1 E0 [X1 ])
R 1 Cov0 [X1 ; T1 ], and
((P + R 1 E0 [X1 ])2 =(1 + R 1 ) + R 1 V ar0 [X1 ]) < 0.
@P =@Corr0 [X1 ; G1 ] < 0.
In general, the sign of @P =@V ar0 [X1 ] depends on Corr0 [X1 ; G1 ]. However, in the common case where Corr0 [X1 ; G1 ] > 0, we have @P =@V ar0 [X1 ] < 0.
Proof. See Appendix B.
The …rst four comparative statics here are similar to those on quantities. If a change in
parameter values implies that the government would like to do more of a project, then if
the supply of the project is …xed at q, price must rise for equilibrium to be reached. The
comparative static with respect to the quantity supplied q is negative. Regardless of the nature
of the program, its price at the margin must fall as the government increases its scale. The
reason for this is that any program, whether it generates positive or negative expected riskadjusted cash ‡ows, increases the volatility of the government budget. Since there are convex
costs of tax distortions, as the size of the program increases, this volatility e¤ect drives up
the expected costs of tax distortions. The price of the program must fall to compensate the
government for these costs. Finally, the government is willing to pay a lower price for programs
whose t = 1 outlays are (i) highly correlated with G1 and (ii) more variable (under regularity
conditions).
3.1.3
Sub-cases
Two special cases of the two-period model are worth considering.
No tax distortions First, consider the case where there are no tax distortions so that
= 0.
In this case, the government can now frictionlessly convert output from Yt to Yt + q (W1
X1 ).
Since we assume that there are no diminishing returns to the project, the optimal solution is
to always go to a corner by either setting q to zero or its maximum feasible value. Speci…cally,
the government will set q to the maximum value if
(W0
X0 ) + R 1 E0 [W1
X1 ] > 0
(W0
X0 ) + R 1 E0 [W1
X1 ] < 0:
and will set q = 0 if
We consider the case of diminishing project returns in one of the extensions below.
22
No net social bene…ts The second special case to consider is one where there are tax
distortions but no net social bene…ts from the project— i.e., where W1
X1 = W 0
X0 = 0.
When the project has no net social bene…ts, the only reason for the government to undertake
it is for tax-hedging motives as in Bohn (1990). Speci…cally, with zero net social bene…ts, the
government’s optimal choice is the weighted average of the solutions in two limiting cases
q
(X0 +R 1 E0 [X1 ])2
1+R 1
(X0 +R 1 E0 [X1 ])2
+ R 1 V ar0
1+R 1
=
R 1 V ar0 [X1 ]
(X0 +R 1 E0 [X1 ])2
1+R 1
!
[X1 ] .
!
+ R 1 V ar0 [X1 ]
First, if the project is riskless (i.e., V ar0 [X1 ] = 0), q =
G0 + R 1 E0 [G1 ]
X0 + R 1 E0 [X1 ]
Cov0 [X1 ; G1 ]
V ar0 [X1 ]
.
(G0 + R 1 E0 [G1 ]) = (X0 + R 1 E0 [X1 ]),
which implies 0 = q (X0 + R 1 E0 [X1 ])+(G0 + R 1 E0 [G1 ]). Assuming that G0 +R 1 E0 [G1 ] >
0, the government wants to undertake projects that generate discounted risk-adjusted inlays
(i.e., X0 + R 1 E0 [X1 ] < 0) in order to lower households overall tax burden. Second, suppose
that X0 + R 1 E0 [X1 ] = 0, so that project outlays are zero NPV. This would be the case if
the project’s cash‡ows represent an asset that is priced by households. In this case, we have
q =
Cov0 [X1 ; G1 ] =V ar0 [X1 ] = arg minq V ar0 [T1 ] = arg minq V ar0 [G1 + qX1 ]. In other
words, the government simply chooses q to minimize the variability of time 1 taxes.
3.1.4
Numerical Example
We next present a simple numerical example to demonstrate that …scal risk can have a quantitatively meaningful e¤ect on the optimal size of a government project. We work with the
two-period model for simplicity. Assume that Y0 = 100 and G0 = 30 so that initial government
spending is 30% of GDP. At t = 1, there are two possible states: a high state, which occurs
with probability 90%, and a low state, which occurs with probability 10%. In the high state,
the stochastic discount factor takes value 0:8, and in the low state it takes value 2:5. These
values imply that the risk-free rate is 3%, and the maximum Sharpe ratio in the economy is
0:525. Assume h (T ) = 0:25% T 2 .
We consider a government program that has the pro…le of a …nancial stability program. We
assume that the program generates slightly negative social returns at t = 0, W0 =
H
t = 1 if the high state is realized, (W1 ) =
0:5, and at
0:5: There are no government cash ‡ows at t = 0
or if the high state is realized at t = 1: X0 = 0 and (X1 )H = 0. If the low state is realized,
the government incurs signi…cant costs (X1 )L =
L
5, but the program generates substantial
social bene…ts (W1 ) = 10. The optimal quantity q of such a program can be interpreted as
23
representing the expansiveness of government protection of the …nancial system. Speci…cally,
larger values of q can be interpreted as the protection of a larger set of …nancial intermediaries
or a larger set of …nancial sector liabilities (e.g., deposits, commercial paper, repo, etc). Note
that we are assuming constant returns to scale for this …nancial stability program.
To emphasize the importance of …scal risk, we consider two paths of other government
expenditure at time 1. In the baseline, government expenditure is acyclical and we have GH
1 =
GL1 = 35. In this case, the optimal size of the …nancial stability program is q = 4:1. This
implies that in the low state, the government is incurring costs of 20:3, or approximately 20%
L
of GDP. In contrast, suppose government expenditure is cyclical with GH
1 = 30 and G1 = 40.
Then the optimal size of the program is q = 2:6, 35% smaller. Thus, the e¤ect of …scal risk is
quite large, even though the realized value of tax distortions is quite small in this example –it
averages 2% of time 0 GDP.
3.2
Risk-Neutral In…nite Horizon Model with AR(1) Dynamics
3.2.1
Set up and Solution
The second special case of the model we consider is one where government spending Gt , project
social bene…ts Wt , and outlays for the project Xt all follow AR (1) processes. In addition, we
assume risk neutrality and h (T ) = ( =2) T 2 for analytical tractability. Relative to the twoperiod model, this parameterization allows us to study the dynamic implications of the model.
Speci…cally, we assume
Let
2
G
= V ar ["G;t ],
2
X
Gt
G =
G
Gt
1
G + "G;t
Xt
X =
X
Xt
1
X + "X;t
Wt
W =
W
Wt
= V ar ["X;t ],
2
W
(20)
W + "W;t .
1
= V ar ["W;t ], and %XG = Corr ["X;t ; "G;t ].
(n)
Since households are risk neutral, the term structure of bonds is trivially Rt
= 1= = R
for all t and n. Thus, without loss of generality, we can assume that the government is only
borrowing short-term, rolling over 1-period debt from one period to the next. Under these
assumptions, optimal taxes are
Tt =
=
RDt
1
+
P1
j
j=0 R Et [Gt+j + qXt+j ]
P1
j
j=0 R
G + qX + (R
1) Dt
24
1
+
Gt
R
G
G
(21)
+q
Xt
R
X
X
:
Naturally, we have
2
and @ Tt =@Gt @
G
@Tt
@Gt
= (R
1) = (R
2
> 0 and @ Tt =@Xt @
< 1 and @Tt =@Xt = q (R
G)
X
1) = (R
X)
q
> 0. Current taxes respond less than one-for-one to
changes in current government outlays. However, they respond more aggressively when shocks
to outlays are more persistent. In the limit where shocks to outlays are a random walk, taxes
must respond one-for-one. Debt evolves as
Dt = RDt
= Dt
so @Dt =@Gt = (1
(R
1) = (R
1
1
+ Gt + qXt
+ Gt
G ) = (R
2
G)
G)
2
(22)
Tt
1
R
G
G
+ q Xt
G
< 1 and @Dt =@Xt = q (1
< 0 and @ Dt =@Xt @
X
1
R
X
=
q (R
X
X ) = (R
2
X)
1) = (R
,
X
X)
< q and @ 2 Dt =@Gt @
G
=
< 0.10 Debt is used to
smooth taxes when there is a shock to government outlays. Current debt responds less than
one-for-one to changes in current government outlays. However, debt responds less aggressively
when shocks to outlays are more persistent. In summary, permanent shocks to outlays have a
large impact on taxes, whereas transitory shocks to outlays are smoothed using borrowing and
have little impact on taxes.
We now characterize the optimal level of q. The optimal level of q satis…es
0=
P1
t=0
R t E0 [Wt
P1
Xt ]
t=0
P1
R t Et [Xt ] T0
t=0
R t Cov0 [X; Tt ] ;
(23)
which is the analog of (15). Suppose we start in the steady state at t = 0 with debt D 1 . Since
we start in the steady state, we have
0=
X G + qX + (R 1) XD
1 R 1
W X
1 R 1
P1
1
t=0
R t Cov0 [Tt ; Xt ] :
Using these expressions, as shown in the Appendix, we can solve directly for the optimal size
of the program q :
q =
W
X
X G + (R
X
where V ar [Xt ] =
2
X = (1
2
X)
2
+
1) D
1
R
2
X
X
1
1
R
R 1
V
R X
ar [Xt ]
G X
R 1
Cov
R G
[Xt ; Gt ]
,
is the unconditional variance of X, and Cov [Xt ; Gt ] = %XG
is the unconditional covariance of X and G.
10
X
Obviously, @Tt =@Gt + @Dt =@Gt = 1 and @Tt =@Xt + @Dt =@Xt = q.
25
(24)
X
X = (1
X G)
3.2.2
Comparative Statics
Proposition 4 In the in…nite-horizon model, assume that W , X, and G follow AR(1) processes,
agents are risk-neutral, and h (T ) = ( =2) T 2 . For simplicity, also assume that q > 0, X > 0,
G > 0,
X
< R 1 , and
@q =@
X
G
< R 1 . We then have the following comparative statics results:
< 0 either holding …xed
2
X
or V ar [Xt ] =
2
G
or V ar [Gt ] =
2
X = (1
2
X ).
@q =@V ar [Xt ] < 0
@q =@X < 0
@q =@
G
< 0 either holding …xed
2
G = (1
2
G ).
@q =@Cov [Xt ; Gt ] < 0
@q =@G < 0.
@q =@ < 0
Proof. Di¤erentiation of equation (24).
The optimal scale of the government program decreases with the mean, variance, and timeseries persistence of the outlays it requires. The optimal scale of the project also decreases with
the mean and time-series persistence of other government spending Gt , as well as the covariance
between other government spending and program outlays. All of these factors increase the
amount and variability of taxes when the government undertakes the project. Since taxes are
distortionary, these factors make the project less attractive to the government, all else equal.
4
Portfolios of Programs
A key feature of our approach is that it easily extends to the case where we have a portfolio of
many government projects. The same basic tradeo¤s between social and …scal risk management
that we derived above will apply here, but they now inherit a portfolio management ‡avor. For
simplicity, we study the optimal portfolio of government programs in the context of the twoperiod model developed above in Section 3.1.
Let projects be indexed by j = 1; :::; J . For t = 0; 1, the government outlays for project j
are Xj;t , social payo¤s are Wj;t ; and the total quantity of project j is qj . Household consumption
is
C t = Yt +
PJ
j=1 qj
(Wj;t
Xj;t )
( =2) (Tt )2
Gt = Yt + (wt
26
xt )0 q
( =2) (Tt )2
Gt (25)
for t = 0; 1, where we use the vector notation [xt ]j =Xj;t , [wt ]j =Wj;t , and [q]j =qj . The
government chooses q and D0 to maximize u (C0 ) + E0 [u (C1 )].
As above, the condition for optimal time 0 borrowing is T0 = E0 [T1 ]. Using the accounting
D0 + G0 + q0 x0 and T1 = RD0 + G1 + q0 x1 , this implies that
de…nitions of taxes T0 =
D0 =
where R = (E0 [M0;1 ])
1
G0
E0 [G1 ] + q0 (x0
1+R
E0 [x1 ])
,
(26)
is the gross interest rate between time 0 and 1. The solution for
optimal borrowing can be substituted into the accounting de…nitions of taxes to obtain
G0 + R 1 E0 [G1 ] + q0 (x0 + R 1 E0 [x1 ])
1+R 1
1
G0 + R G1 + q0 (x0 + R 1 x1 ) + (G1 E0 [G1 ]) + q0 (x1
=
1+R 1
(27)
T0 =
T1
E0 [x1 ])
,
(28)
which clearly satisfy the optimality condition for debt and taxes since T0 = E0 [T1 ].
Thus, the system of …rst order conditions that de…ne the optimal scale of these projects is
given by
0 = (w0
x0 (1 + T0 )) + R 1 E0 [(w1
(29)
x1 (1 + T1 ))] ;
which is simply the vector analog of equation (9) above. Plugging in the solutions for optimal
taxes and solving for q, we obtain
1
=
q
|
(x0 + R 1 E0 [x1 ])(x0 + R 1 E0 [x1 ])0
+ R 1 V ar0 [x1 ]
1+R 1
{z
1
Portfolio weights absent …scal hedging motives
(x0 + R 1 E0 [x1 ])(x0 + R 1 E0 [x1 ])0
+ R 1 V ar0 [x1 ]
1+R 1
|
{z
1
"
(w0 x0 ) + R 1 E0 [w1 x(30)
1]
}
(G0 +R
1 E [G ])(x +R 1 E [x ])
1
0
0
0 1
1+R 1
+R 1 Cov0 [x1 ; G1 ]
Fiscal hedging motive
which is just the natural vector analog of equation (19) above.
4.1
Application: Ex-ante regulation versus ex-post bailouts
We can use this setup to think about the desirability of ex-ante regulations versus ex-post
bailouts for …nancial stability, sometimes called the “lean vs. clean”debate.11 Both regulation
and bailouts can have large …nancial stability bene…ts because they help to mitigate severe
…nancial crises. Regulation can be used to lean against asset bubbles ex-ante, reducing the
11
http://www.federalreserve.gov/newsevents/speech/stein20131018a.htm
27
#
}
;
probability and severity of …nancial crises. But this can come at the cost of ine¢ ciently reducing economic growth if regulation dampens welfare-improving innovations or unnecessarily
constricts credit supply. Alternatively, the government can use bailouts to clean up problems ex
post. Ex-post bailouts leave ex-ante growth unfettered but require a larger use of government
balance sheet capacity in the event of a …nancial crisis. A key point that emerges from our
framework is that the relative quantity of each intervention will vary with the degree of tax
distortions and the existing government debt burden.12
To examine this formally in our framework, consider the two-period (t = 0; 1) version of the
model. Denote ex-ante interventions with j = 1 and ex-post bailouts with j = 2. Suppose that
Wj;0 = 0 and Xj;0 = 0 for j = 1; 2 in order to focus on period 1 cash ‡ows.13 Suppose that
ex-ante regulation (j = 1) requires the government to incur some small constant costs through
its government balance sheet so that we have constant X1;1 = k > 0. These costs can be
thought of as the costs of paying regulators and conducting bank examinations.
Given the assumptions, the optimal choices are given by
" #
q1
q2
1+R
=
k
E0 [W1;1 ]
k
1 E0 [X2;1 ]
+
k V ar0 [X2;1 ]
(G0 + R
1
1
" #
1
0
E0 [W2;1 ]
E0 [X2;1 ]
" #
E0 [g1 ])
R
k
0
E0 [W1;1 ]
k
"
Cov0 [g1 ; X2;1 ]
V ar0 [X2;1 ]
#
E0 [X2;1 ]
"
k
E0 [X2;1 ]
k
1
#
.
This formula illustrates the forces that determine q1 and q2 . First, an increase in the riskadjusted e¢ cacy of ex-ante regulation, E0 [W1;1 ] =k, increases q1 . Second, an increase in the differential e¢ cacy of ex-post bailouts versus ex-ante regulation, E0 [W2;1 ] =E0 [X2;1 ] E0 [W1;1 ] =k,
leads the planner to substitute from ex-ante regulation towards ex-post bailouts. Speci…cally,
the government will only choose q2 > 0 if E0 [W2;1 ] =E0 [X2;1 ] > E0 [W1;1 ] =k— i.e., if the riskadjusted bang-for-buck is greater for ex-post interventions. Given the desire the smooth taxes,
this substitution is greater when V ar0 [X2;1 ] is small. The …nal two terms capture the way that
background …scal risk impact the choices of q1 and q2 .
The following proposition describes the behavior of the optimum. In order to consider the
12
de Faria e Castro, Martinez, and Philippon (2014) also study the relationship between …scal capacity and
…nancial stability interventions. However, their focus is on government policies that disclose information about
asset quality in the …nancial sector.
13
This is without loss of generality since, as show above in (30), all that matters for j = 1; 2 are the NPV
of net bene…ts minus outlays ((w0 x0 ) + Rf 1 E0 [w1 x1 ]), the NPV of outlays (x0 + Rf 1 E0 [x1 ]), time 1
variance of outlays (V ar0 [x1 ]), and covariance of outlays with background spending (Cov0 [x1 ; G1 ]).
28
e¤ects of scaling all other government expenditure up or down, we write G1 = g1 and compute
comparative statics with respect to .
Proposition 5 Suppose that ex-ante regulation and ex-post bailouts have the cash ‡ow characteristics assumed above. We have the following comparative statics:
@q1 =@G0 =
R=k < 0
@q2 =@G0 = 0,
1
(E0 [g1 ]
E0 [X2;1 ] (Cov0 [X2;1 ; G1 ] =V ar0 [X2;1 ]))
@q1 =@ =
k
@q2 =@ =
Cov0 [g1 ; X2;1 ] =V ar0 [X2;1 ], and
@q1 =@ =
f(1 + R) = ( 2 k)g fE0 [W1;1 ] =k
1g
+ (E0 [X2;1 ])2 = (V ar0 [X2;1 ] ( 2 k)) fE0 [W2;1 ] =E0 [X2;1 ]
@q2 =@ =
E0 [X2;1 ] = ( k)2 V ar0 [X2;1 ]
E0 [W1;1 ] =kg
fE0 [W2;1 ] =E0 [X2;1 ]
E0 [W1;1 ] =kg
The comparative statics have strong analogies to portfolio choice logic. As the level of initial
government expenditures G0 rises, all adjustment takes place by reducing the amount of ex-ante
regulation. Since it always incurs constant cost k, regulation is essentially a riskless technology,
so when riskless government expenditures G0 go up, all the adjustment comes from reducing
regulation. This is analogous to the portfolio choice logic that dictates that with CARA utility,
the total dollar amount invested in risky technologies is constant in total wealth. Only the
amount invested in riskless technologies varies.
When the amount of government expenditure at time 1, G1 = g1 , is scaled up, the comparative statics depend on Cov0 [G1 ; X2;1 ], the (risk-adjusted) covariance between other government
expenditures and bailout cash ‡ows. It is natural to assume that Cov0 [G1 ; X2;1 ] > 0 since the
government is likely to be spending a lot on other expenditures like unemployment bene…ts
(high G1 ) at times when it is making bailout payments (i.e., when X2;1 is high). In this case,
we have @q2 =@ < 0. Raising other government expenditures makes ex-post interventions less
attractive because they are correlated with other government budget expenditures. This raises
the government’s total tax burden and thus the total distortions incurred. As in portfolio choice
problems, the degree of adjustment in the quantity of bailouts q2 depends on the (risk-adjusted)
beta of government expenditures with respect to bailout payo¤s.
When the amount of government expenditure G1 = g1 rises, there are two e¤ects on the
quantity of regulation, q1 . To see the …rst e¤ect, suppose E0 [g1 ] = 0. Then
@q1
E [X2;1 ] Cov0 [X2;1 ; G1 ]
= 0
> 0:
@
k
V ar0 [X2;1 ]
29
Thus, at low levels of other government expenditures, increases in
make bailouts less attrac-
tive, so we substitute towards ex-ante regulation. However, there is a second countervailing
e¤ect when E0 [g1 ] > 0. When the expected tax burden is too high, increases in
make all
government expenditure less attractive. If this e¤ect overwhelms the substitution e¤ect, it may
be optimal to cut back on both ex-post bailouts and ex-ante regulation as
rises.
These results can also be interpreted as motivating "…nancial repression" at high levels
of government debt. Reinhart and Sbrancia (2011) argue that at high levels of government
debt, …nancial regulation is used to force …nancial intermediaries to hold government debt,
providing a captive buyer. Our model makes the point that …nancial repression may be optimal
in high debt situations, not just because the government needs a buyer of debt but because the
government cannot a¤ord the costs of the alternative …nancial stability policy –bailouts.
Similar logic applies to the comparative statics on the severity of tax distortions . We have
@q1
@
@q2
@
=
=
1 1 + R E0 [W1;1 ]
1 (E0 [X2;1 ])2
1 + 2
2
k
k
k V ar0 [X2;1 ]
1 E0 [X2;1 ]
E0 [W2;1 ] E0 [W1;1 ]
:
2 V ar [X ]
E0 [X2;1 ]
k
2;1
0
E0 [W2;1 ]
E0 [X2;1 ]
E0 [W1;1 ]
k
To the extent that ex-post bailouts are highly attractive relative to ex-ante regulation on a
risk-adjusted basis (i.e., E0 [W2;1 ] =E0 [X2;1 ]
E0 [W1;1 ] =k is large), an increase in distortions
pushes the government away from ex-post bailouts and towards ex-ante regulation (i.e., we
have @q2 =@ < 0 and @q1 =@ > 0). However, to the extent that ex-ante regulation is highly
attractive (i.e., E0 [W1;1 ] =k
1 is large), there is an o¤setting e¤ect: in this case regulation
expenditures are already quite large, so the increase in distortions leads the government to cut
back on regulation as well.
We have discussed comparative statics in terms of quantities here. As pointed out above,
if we …x the quantities of regulation and bailouts, comparative statics in terms of their prices
will simply have the opposite signs. Thus, the prices charged for bailouts should rise as other
government expenditures
4.2
and tax distortions
increase.
Application: Optimal budget cuts
We can use the same basic setup to study how the government should react to …scal shocks.
Consider two government projects. The …rst, denoted with j = 1, has expenditures that do not
covary with other government programs. Spending by the National Science Foundation might
be an example of such a project. As above, we model this project as having constant time 1
expenditures X1;1 = k: The second project, denoted j = 2, has expenditures that do covary
30
with other government spending. Spending on automatic stabilizers would be an example of
such a project. We denote these expenditures X2;1 .
Our model highlights two di¤erent types of …scal shocks, which the government should
respond to di¤erently. First, there could be an unexpected shock at time 0, which raises
government spending at that time, G0 , without a¤ecting expectations of the future. As shown
by the comparative statics derived in Proposition 5, @q1 =@G0 and @q2 =@G0 , such shocks should
be fully adjusted to by reducing the riskless project j = 1. The intuition here is that the
government should try to o¤set one-time shocks rather than incur additional tax distortions by
adjusting the overall path of expenditures.
In contrast, suppose the …scal shock increases expectations of future government expenditure. Then the comparative statics @q1 =@ and @q2 =@ in Proposition 5 show that there are
both income and substitution e¤ects. The government should substitute towards projects with
lower …scal risk, i.e., those that covary less with future government expenditure. Income e¤ects
push towards shrinking all programs.
5
5.1
Extensions
Nonlinear Government Technology
In the main modelm we assume that there are constant returns to scale in the projects the
government is considering. Of course, in reality there may be increasing or decreasing returns to
scale for di¤erent projects undertaken by the government. This idea can be easily incorporated
into our basic framework by assuming that the social returns to a project of scale q are f (q) Wt
rather than qWt , where f 0 > 0 and f 00 < 0 in the diminishing returns case and f 00 > 0 in the
increasing returns case. Similarly, we assume that outlays are g (q) Xt rather than qXt where
g 0 > 0 and g 00 > 0 in the diminishing returns case and g 00 < 0 in the increasing returns case.
Under these assumptions, household consumption is
Ct = Yt + f (q) Wt
g (q) Xt
( =2) (Tt )2
Gt
and the optimal amount of government activity, q , now satis…es
0 =
P1
(t) t
t=0 (R0 ) E0
g 0 (q )
g 0 (q )
[f 0 (q ) Wt
P1
(t) t
t=0 (R0 ) E0
P1
(31)
[Xt ] T0
(t) t
t=0 (R0 ) Cov0
31
g 0 (q ) Xt ]
[Xt ; Tt ] :
(32)
Obviously, this expression collapses to (15) when f (q) = g (q) = q.
For instance, one can use this extension to think about the Powell doctrine for government …nancial stability interventions. The Powell doctrine is the idea that it is optimal to use
overwhelming force when dealing with …nancial crises. A powerful show of government intervention early in a crisis can restore con…dence and prevent ampli…cation mechanisms like …re
sales from taking hold. In our setup, one could think of representing this idea with increasing
returns to scale for …nancial bailouts over some region. Speci…cally, if f 00 > 0 over some region
then it could be the case that small interventions are negative NPV, so that evaluating the
government’s …rst order condition at 0 we have
0>
P1
(t) t
t=0 (R0 ) E0
[f 0 (0) Wt
g 0 (0) Xt (1 + Tt )] :
However, since f 00 > 0, it can still be the case that the optimal scale is q > 0.
It may also be interesting to use our setup to think about how the Powell doctrine interacts
with uncertainty about the necessary size of the bailout. Increasing returns to scale make the
case for a large intervention, but if the size of the …nancial sector’s capital hole is uncertain,
there is the risk of ending up like Ireland instead of the US in the recent …nancial crisis. In
the language of the model, one would be trading o¤ the increasing returns to scale to bailouts
against increasing expected distortionary costs.
A weaker version of the Powell doctrine that may apply more generally is that government
ine¢ ciency may raise the hurdle rate for government interventions.
5.2
Discretionary Policies
We now consider the case of a discretionary …scal policy that can be varied over time. Speci…cally, we assume that at time t the government chooses the size of the program that will prevail
at time t + 1. However, the relevant realizations of fWt+1 ; Xt+1 ; Gt+1 g are not known for certain
at time t when this policy is chosen. For simplicity, we assume that households are risk-neutral
and that h (T ) = ( =2) T 2 . The government’s problem is
max
fDt ;qt g
P1
j=0
R t E 0 qt
1
(Wt
Xt )
( =2) (Gt + RDt
1
+ qt 1 Xt
Dt )2 ,
(33)
taking the path of fWt ; Xt ; Gt g as given.
The …rst order condition for qt is
Et [Wt+1
Xt+1 ] = Et [Xt+1 Tt+1 ]
32
(34)
and the …rst order condition for Dt implies that Tt = Et [Tt+j ] for all j > 0. The lifetime
P
j
government budget constraint as of time t is RDt 1 = 1
Gt+j qt+j 1 Xt+j ].
j=0 R Et [Tt+j
Combining this with the fact that Tt = Et [Tt+j ] implies that optimal taxes at time t are
Tt = (R
1) Dt
1
+
R
R
1 P1
j=0
R j Et [Gt+j + qt+j 1 Xt+j ] .
Plugging this solution for Tt into Dt = Gt + qt 1 Xt + RDt
1
(35)
Tt and rearranging, we see that
taxes tomorrow equal taxes today plus the innovation to expectations of lifetime spending
Tt+1 = Tt + (Et+1
Et )
R
R
which implies that
qt
1
=
1 P1
j=0
R
j
(Gt+1+j + qt+j Xt+j+1 ) ;
R
Et [Wt+1 Xt+1 ]
R 1 V art [Xt+1 ] + R 1 (Et [Xt+1 ])2
P
j
Et [Xt+1 ] RDt 1 + 1
j=0 R Et [Gt+j ] + qt 1 Xt + R
(37)
1
P1
j=1
R j Et [qt+j Xt+j+1 ]
V art [Xt+1 ] + R 1 (Et [Xt+1 ])2
P1
P1
j
j
j=1 R Covt [Xt+1 ; qt+j Xt+j+1 ]
j=0 R Covt [Xt+1 ; Gt+1+j ] +
V art [Xt+1 ] + R
1
(Et [Xt+1 ])2
(36)
.
Thus, assuming that qt > 0, the discretionary program is large today when:
Et [Wt+1
Xt+1 ] is large, so the project is expected to generate large net social bene…ts
next period.
V art [Xt+1 ] + R
1
(Et [Xt+1 ])2 is small, so the project adds little to either the expected
level or volatility of taxes tomorrow.
The sum of past accumulated debt, expected future non-discretionary spending, past discretionary spending, or expected future discretionary spending are low. Each of these
reduces Et [Tt+1 ] for a given choice of qt which makes discretionary spending more attractive in the presence of distortionary taxes.
P1
j=0
R j Covt [Xt+1 ; Gt+1+j ] is small so shocks to discretionary spending at t + 1 are not
strongly correlated with news about future non-discretionary spending.
P1
j=1
R j Covt [Xt+1 ; qt+j Xt+j+1 ] is small so shocks to discretionary spending at t + 1 are
not strongly correlated with news about future discretionary spending.
33
6
Conclusion
We present a model in which the distortions associated with taxation make …nancing costly
for the government. We explore the consequences of this assumption for the set of programs
that the government chooses to undertake. As in the literature on costly external …nance in
corporate …nance, we show that distortionary taxation impacts the optimal scale and pricing
of government programs. In particular, the government has both social and …scal risk management motives. The social risk management motive arises from the fact that some government
programs deliver large bene…ts when marginal utility is high. The …scal risk management motive arises from the government’s desire to avoid raising distortionary taxes both on average
and in particular when marginal utility is high. Fiscal risk gives the government a hedging
motive and means that individual programs cannot be evaluated in isolation – the …scal risk
of a particular program depends on the other programs and expenditures the government is
undertaking.
We highlight the interaction between the social and …scal risk management motives. These
motives may con‡ict if programs with strong social risk management bene…ts involve government expenditures and hence higher tax distortions in bad times, creating more …scal risk.
On the other hand, they may reinforce each other if social risk management preserves private
output in a downturn— reducing the distortionary impact of a given dollar amount of taxes.
34
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38
A
Mapping Examples into the Model Setup
To make the setup concrete, we present two simple examples.
A.1
Financial stabilization
The …rst is an example from …nance that considers the value of government interventions to
prevent …re sales. We take the Stein (2012) model of …re sales as our starting point. In
the model, there are three periods t = 0; 1; 2: Levered intermediaries hold assets I and back
fraction d with riskless short-term debt. Bad news at the interim date (t = 1) means that there
is probability (1 q) that the assets are worth zero at t = 2 and probability q that they are
worth I=q at t = 2. Thus, if there is bad news at the interim date, the assets are worth I
in expectation. To render the short term debt riskless, intermediaries must ensure that the …re
sale value of the assets at the interim date can cover repayment of the debt. That is, we must
have
dI = Rf 1 k I
where 0 < k 1 is the …re sale discount at the interim date. The discount is pinned down by
the capital W and outside opportunities g ( ) of patient investors, who will purchase assets …re
sold by levered intermediaries at t = 1. Speci…cally, we have
1
= g 0 (W
k
dIRf ) :
The real cost of the …re sale is that patient investors only invest W dIRf in new outside
projects rather than W if the …re sale takes place.
Now suppose there is bad news at t = 1 so that there is a …re sale in the absence of
government intervention. Consider a government program where the government steps in and
guarantees the short-term debt . For simplicity, think of the government as assuming all the
assets and liabilities of the intermediaries and guaranteeing their short-term debt so that it does
not run at the interim date. This assumption ensures that there are no in‡ows or out‡ows of
cash at t = 1. Further assume the guarantee is unanticipated so that it has no ex-ante e¤ects.
However, this guarantee prevents the …re sale from taking place at t = 1.
In our notational conventions, the government’s outlays at t = 2 are X2 = dI if the outcome
is bad, and the assets are worth zero (probability 1 q) or X2 = dI
I=q if the outcome is
good and the assets recover (probability q). The externality cash ‡ow from this government
program is the gain in net output relative to the counterfactual with no government guarantee
W1 = [g (W )
A.2
W]
[g (W
dIRf )
(W
dIRf )] :
Automatic stabilizers
Our second example considers unemployment insurance, taking the Rothschild-Stiglitz (1976)
model as our baseline.14 There are two types of individuals, those (L-types) with low probability
of being unemployed pL and those (H-types) with high probability of being unemployed pH .
14
As we discuss further below, there is some tension in our use of a representative agent framework in situations
where there are multiple types of agents.
39
Agents know their types, and fraction f are high types, and (1 f ) are low types. Both types
earn income E1 when they are employed and income E2 < E1 when they are unemployed. The
insurance contract speci…es that the agent pays the insurer 1 if he is employed and receives 2
if unemployed. With insurance premia set by risk-neutral intermediaries, insurance is actuarily
fair, and we have
1 p
1
2 =
p
where p is the probability of being unemployed by those agents who purchase insurance. It is
well-known that no pooling equilibrium exists in this setting.
The separating equilibrium that can survive is one where high types fully insure so their
marginal utility is equated across states
u0 (E1
1;H )
= u0 E2 +
1
pH
;
1;H
pH
and low types receive so little insurance that the high types do not wish to deviate:
(1
pH ) u (E1
1;H )+pH u
E2 +
1
pH
(1
1;H
pH
pH ) u (E1
1;L )+pH u
Expected welfare in the private market separating equilibrium is
2
Wpriv = (1
6
6
f ) 6(1
4
|
2
6
6
+f 6(1
4
|
pL ) u E 1
1;L
+ pL u E 2 +
{z
1
1;L
}
EU for L-types
pH ) u E1
+ pH u E 2 +
{z
1;H
1
3
pH
pH
EU for H-types
1;H
}
1
pL
pL
3
pL
pL
E2 +
7
7
7
5
7
7
7:
5
Suppose the government mandates insurance where the employed pay 1 and the unemployed receive 2 . Furthermore, we now introduce aggregate risk by having p, the average
probability of being unemployed, vary over time. We assume that 1 and 2 do not vary with
p, so the net government expenditure associated with unemployment insurance will vary as a
function of p. Expected welfare is
Wgovt = (1
p) u (E1
1
) + pu (E2 +
2
);
so that
W1 = Wgovt
Wpriv :
And net government outlays are
X1 = p
which rises with p so long as
(1
2
2
>
1
p)
1
= p(
2
1
)
1
;
. In this context, assuming a broad program of unem40
1;L
:
ployment insurance with a given level of bene…ts
optimal employment insurance 1 .
B
2,
one could use our program to solve for the
Omitted Proofs
B.1
Proof of Proposition 3
The Euler equation de…ning P is
P (1 + T0 ) + R 1 E0 [W1
0 = f (P ; ) = W0
X1 (1 + T1 )] :
For any parameter ;comparative statics follow from
0 = fP (P ; )
)
Since
@P
=
@
@P
+ f (P ; )
@
f (P ; )
:
fP (P ; )
@T1
@T0
q
=
=
@P
@P
1+R
we have
fP (P ; ) =
1+
1
;
G0 + R 1 E0 [G1 ] + 2q (P + R 1 E0 [X1 ])
1+R 1
<0
In a neighborhood of zero-subsidy price P + R 1 E0 [X1 X1 ] = 0, we always have fP (P ; ) < 0.
Thus, we will assume that fP (P ; ) < 0 so that we have
@P
=
@
f (P ; )
/ f (P ; ) .
fP (P ; )
1. E¤ect of : We have
@P
/
@
T0 P + R 1 E0 [X1 ]
R 1 Cov0 [X1 ; T1 ] :
2. E¤ect of G0 : We have
@P
/
@G0
1+R
1
P + R 1 E0 [X1 ] :
3. E¤ect of W0 and E0 [W1 ]: We have
@P
@P
> 0 and
> 0.
@W0
@ (E0 [W1 ])
41
4. E¤ect of q: We have
2
(P + R 1 E0 [X1 ])
+ R 1 V ar0 [X1 ]
1+R 1
@P
/ fq (P ; q) =
@q
B.2
!
< 0.
Proof of Proposition 4
Recall that
Tt = G + qX + (R
and
Dt = Dt
1
+ Gt
1) Dt
1
R
G
1
G
G
Gt
R
+
+q
G
+ q Xt
Xt
R
X
X
1
R
X
G
X
.
X
Since we start in the steady state at t = 0 with debt D 1 , we have
P1
X G + qX + (R 1) XD 1
t
t=0 R Cov0 [Tt ; Xt ] :
1
1 R
P
t
Thus, to solve for q , we simply need to evaluate 1
t=0 R Cov0 [Tt ; Xt ].
One subtle point here is that the optimal level of taxes at time t depends on the amount
of accumulated debt Dt 1 . And the amount of accumulated debt is a random variable which
depends on the past
P1path of t Xt and Gt . We need to take this path-dependency into account
when computing t=0 (Rf ) Cov0 [Tt ; Xt ]. Fortunately, this is easy given the assumed AR (1)
dynamics. Speci…cally, accumulated debt at t 1 is
0=
Dt
1
W X
1 R 1
=D
1
+
1
R
Pt
1
j=1
G
G
so we have
Tt =
which implies
Gt
1
R
G +q
j
G + qX + (R 1) D 1
Pt 1
R 1
+
(1
G)
j=1 Gt j
R
G
Pt 1
R 1
+q
(1
X)
j=1 Xt
R
X
Cov0 [Tt ; Xt ] =
R
R
+q
1
(1
G)
G
R
R
1
(1
Pt
1
j=1
X)
X
42
X
X
Pt
1
j=1
G + Gt
j
X + Xt
Xt
X ,
j
G
X
;
Cov0 [Gt j ; Xt ] + Cov0 [Xt ; Gt ]
Pt
1
j=1
Cov0 [Xt j ; Xt ] + V ar0 [Xt ] :
Now since Xt = X +
Pt
1
j=0
j
X "X;t j
V ar0 [Xt ] =
1
Cov0 [Xt ; Gt ] =
1
j
XV
Cov0 [Xt j ; Xt ] =
2
X
2 t
X
(
1
t
G X)
j
X Cov0
Cov0 [Gt j ; Xt ] =
and Gt = G +
Pt
1
j=0
j
G "G;t j ,
we have
2
X
%XG G X
1
G X
j
X
[Gt j ; Xt j ] =
j
X
ar0 [Xt j ] =
1
(
1
%XG
1
t j
2
X
t j
2
X
1
G X)
2
X
G X
G X
.
Combining these facts we have
Cov0 [Tt ; Xt ] =
R
R
1
%XG G X
(1
1
G X
2
1
X
(1
2
1
X
X
R 1 1
R
1
G
R 1 1
R
1
G
G
+q
R
R
=
1
t
X
=
1
t
X
Pt
1
j=1
G)
X)
Pt
1
j=1
j
X
1
j
X
(
t j
G X)
2 t j
X
1
+ 1
(
t
G X)
2 t
X
+ 1
2
R 1
X
(1 + X )
2
R
1
X
G X
X
X
R
1
G X
Cov [Xt ; Gt ] + q
(1 + X ) V ar [Xt ]
R
X
X
G X
%XG
1
G X
+q
2
where V ar [Xt ] = 2X = (1
X ) is the unconditional variance of X and Cov [Xt ; Gt ] = %XG
is the unconditional covariance of X and G.Thus, we have
P1
t=0
R t Cov0 [Tt ; Xt ] =
R
R
1
1R
R
R
X
X
1
1
G X
1
G
Cov [Xt ; Gt ] + q
X
R
R
1
X
X = (1
(1 +
X
which implies that the optimal value of q satis…es
q =
W
X
X G + (R
X
B.3
2
+
1) D
1
R
2
X
X
1
1
R
R 1
V
R X
ar [Xt ]
G X
X
R 1
Cov
R G
[Xt ; Gt ]
.
Proof of Proposition 5
Suppose that w0 = x0 = 0, G1 = g1 , and X11 = k. Then the general expression for q in the
two-period model reduces to
q1
q2
=
1+R
k
E0 [W1;1 ]
k
(G0 + R
1
1 E0 [X2;1 ]
1
+
0
k V ar0 [X2;1 ]
R
Cov0 [g1 ; X2;1 ]
E0 [g1 ]) k
0
V ar0 [X2;1 ]
1
43
E0 [W2;1 ]
E0 [X2;1 ]
E0 [X2;1 ]
k
1
E0 [W1;1 ]
k
.
E0 [X2;1 ]
k
X) V
X G)
ar [Xt ]
Note that we then have
@q1 =@G0
@q2 =@G0
R
k
=
0
@q1 =@
@q2 =@
=
"
@q1 =@
@q2 =@
=
1+R
2k
E0 [g1 ] =k
Cov0 [g1 ;X2;1 ]
E0
V ar0 [X2;1 ]
Cov0 [g1 ;X2;1 ]
V ar0 [X2;1 ]
E0 [W1;1 ]
k
1
[X2;1 ] =k
#
1 E0 [X2;1 ]
2 k V ar [X ]
2;1
0
1
0
44
E0 [W2;1 ]
E0 [X2;1 ]
E0 [W1;1 ]
k
E0 [X2;1 ]
k