Fiscal Policy, Sovereign Debt and Default
with Model Misspecification
Viktor Tsyrennikov∗
Cornell University
February 5, 2014
Abstract
We study a small open economy model with production and public good provision. The government can borrow from abroad but it
is subject to a premium reflecting probability of default. We show
that when both creditors and the country-debtor have fears of model
misspecification level and volatility of the sovereign bond premium
increase significantly. This stems from the endogenous disagreement
between the agents taking opposite sides of financial trade. Pessimism
stemming from model misspecification leads the government to use
taxes more actively. Out of precaution the government taxes more
in good times and “manages the adverse states” by increasing taxes
heavily. Default periods are associated with a significant increase of
government’s savings as observed in the data.
Keywords: sovereign default, debt, fiscal policy
∗
email: vt68@cornell.edu. I would like to thank ...
1
1
Introduction
We study endogenous fiscal policy and default in a small open economy.
The economy is populated by a representative household that consumes and
supplies labor to the market. The household cannot borrow and all borrowing
is done by the government. The government, in turn, chooses labor tax,
public good provision, and pubic debt to maximize society’s life-time utility.
Government budget does not have to be balanced every period and deficits
(or surpluses) can be financed by borrowing from (or lent to) international
creditors. But when the circumstances lead the government to accumulate
significant debt it may choose to default. The creditors anticipate that the
government may repudiate its debt and price in a possibility of sovereign
default.
Unlike Arellano [2008] and Aguiar and Gopinath [2006] we study a production economy with an endogenously determined fiscal policy. In section 2
we argue that government’s action’s, in particular deteriorating budget balance was one of major reasons for default in Argentina in 2001. Default in
this model is driven by a government’s reluctance to increase taxes or decrease public spending. Government’s taxes are a great policy tool in normal
times. But when productivity is low higher taxes add to default worries by
reducing output while increasing tax revenue only marginally. Provision of
public goods is reduced gradually at times of distress to maintain households’
welfare and not to add to fears of default.
Both households in the country-borrower and creditors are assumed to
have recursive utilities. We study a special case with robust (multiplier)
preferences as in Hansen and Sargent [2007]. This preference specification
locks in the intertemporal elasticity of substitution (IES) at 1 but allows varying the risk-aversion parameter that we denote by θb . Hansen and Sargent
interpret θb as a measure of concern for model mis-specification. Consider
the following thought experiment. An agent believes that distribution F is
a reasonable description of data. But he realizes that there are other choices
F̃ that could also fit the data reasonably well. An agent with rational expectations behaves as if F is the true model. An agent with concerns for model
mis-specification behaves as if the true model is the worst possible assign-
2
ment among possible F̃ . The larger is θb the more different F and F̃ are.
Distribution F̃ is a “pessimistic” version of F . That is it assigns more weight
to adverse, from the agent’s point of view, outcomes. Creditors’ pessimism
increases the country’s borrowing cost by a “pessimism” premium.
We would like to highlight that preference for mis-specification is twosided. This leads to endogenous disagreement between the governmentborrower and creditors who are on the opposite sides of sovereign debt transactions. The endogenous disagreement increases volatility of bond returns.
Disagreement is present even if only one of the agents has concerns for model
mis-specification. Consider the case with pessimistic creditors and a rational
borrower. Because of the payoff structure of the defaultable sovereign debt1
disagreement is only about the “far” left tail of the distribution. When lowest
productivity states occur premia react dramatically. But these states occur
only infrequently; so, volatility of premia increases only marginally. Next,
consider the case with rational creditors and a pessimistic borrower. Bond
prices are closer to beliefs and valuation of the wealthier market participant.
So, as wealth moves between creditors and the borrower bond prices and
returns change significantly.
In a closely related work Presno and Pouzo (2013) extend the analysis in
Arellano [2008] and assume that creditors, but not borrowers, have concerns
about model mis-specification. Pessimistic creditors behave in many ways
like impatient agents. To achieve a reasonable default probability households have to be made unrealistically impatient. With two-sided concern for
model mis-specification the model performs well even with small differences
in discount factors.2 Adam and Grill [2011] study default decisions in the
presence of disaster events. Disaster events, like preference for robustness,
generate caution on the part of both creditors and borrowers. It is another
way of introducing (two-sided) pessimism into economic behavior.
Aguiar and Amador [2011] also study a model with production and default. In their model the government, for political reasons, overvalues current
1
It pays only in high productivity states.
When creditors are pessimistic the borrower has to be made very impatient so that
he would default in equilibrium. When both creditors and the borrower are pessimistic
the borrower would default even if he were relatively patient.
2
3
spending. While the government cannot commit to repaying its obligations
(endogenously incomplete markets) default does not occur in equilibrium. In
this way our model is more suitable for quantitative analysis. But it would be
fruitful to incorporate the political economy frictions into the current model.
Pouzo [2013] also studies optimal fiscal policy with production and default
under incomplete markets. It builds upon (a closed economy of) Aiyagari
et al. [2002] but relaxes the assumption that the government can commit
to a tax and debt policy. The government may choose to repay its debt
only partially. Default does arise in equilibrium but it is a levi on domestic
households rather than foreign creditors.
In what follows we present data facts about the 2001 default episode in
Argentina to justify some of the modelling choices that we make. Description
of the model follows. Then we present our numerical results and conclude.
2
Data facts
Most of the research on sovereign crises concentrates on the three facts. First,
output and consumption are very volatile and strongly correlated. The two
series are plotted in figure 1 panel A. Second, trade balance and current
account (as shares of GDP) are strongly counter-cyclical. These series are
plotted in figure 4 panel B. Third, cost of borrowing for emerging markets is
high and extremely volatile.
While the first two facts can be rationalized within an exchange economy
with incomplete markets and default as in Arellano [2008] and Aguiar and
Gopinath [2006]. But these models are unsuccessful in matching the facts
about country premia. This work aims to provide a rationale for the observed
country premia.
We start by pointing out that borrowing in emerging market economies is
done primarily by a government not private agents. We illustrate this point
using the data on Argentina. Figure 2 panel A shows outstanding public
and private liabilities as percentages of Argentina’s GDP. Private liabilities
are a small fraction, less than 20%, of the total liabilities. Panel B of the
same figure shows net private borrowing from offshore banks. Apart from
4
A. Output and consumption
B. Cost of borrowing - USD LIBOR
0.15
0.80
EMBI Argentina
0.70
0.10
0.60
0.05
0.50
0.00
0.40
0.30
-0.05
0.20
-0.10
output
consumption
0.10
-0.15
0.00
1992 1994 1996 1998 2000 2002 2004 2006
1996
1998
2000
2002
2004
2006
Figure 1: Argentina’s output, consumption and cost of borrowing
A. Outstanding debt, % of GDP
B. Private borrowing, % of GDP
80
70
8
public
private
6
60
4
50
40
2
30
0
20
-2
10
0
-4
92 94 96 98 00 02 04 06 08 10 12
92 94 96 98 00 02 04 06 08 10 12
Figure 2: Argentina’s external liabilities
years 1992-3 it is less than 2% of GDP and it is close to zero on average.
This is a consequence of heavy capital controls used to aid the fixed exchange
5
rate regime. These two facts motivate our assumption that households cannot borrow or lend. That is all the borrowing is done by the government.
Modelling government directly opens a possibility for quantitative analysis
of political risk as in Aguiar and Amador [2011].
A. Government budget, % of GDP
B. External balance, % of GDP
8
6
20
fiscal deficit
interest payments
current account
15
4
10
2
5
0
0
-2
-5
-4
-10
92 94 96 98 00 02 04 06 08 10 12
92 94 96 98 00 02 04 06 08 10 12
Figure 3: Argentina’s budget and current account balances
Further, reversals in budget balance can alone explain a large fraction of
current account reversals in emerging economies.3 Figure 4 panel A plots
the difference between the Argentina government’s revenues and expenditures. This does not include interest payments; this item is plotted separately. Thus at the time of 2001 default government deficit was more than
3% of GDP; at the same time the government paid 4% of GDP in the form of
interest payments. Figure 4 panel B plots the trade and the current account
balances. They increased 13.7% and 10.0% respectively between 2001 and
2002. Current account and trade balance in Argentina are strongly countercyclical. That is lending evaporates at distress times.
3
Ferrero [2010] shows that the U.S. current account can be well explained by the U.S.
fiscal policy stance and demographical changes.
6
Government budget deficit, % of GDP
9
8
IMF calculations
Implied by debt accumulation
7
6
5
4
3
2
1
0
1992
1994
1996
1998
2000
2002
Figure 4: Argentina’s budget and current account balances
3
3.1
Model
Uncertainty, goods and technology
The state zt ∈ S is a first-order Markov process. The state zt represents
productivity of the country borrower. Let dF (zt+1 |zt ) denote the transition
density from state zt into state zt+1 . We denote history of the state up to
date t by z t = (z0 , z1 , ..., zt ).
At each date a consumption good is traded. It is produced using a linear
technology with labor being its single input:
y(zt , lt ) = zt lt .
3.2
(1)
Households
A country-borrower is populated by a representative household with recursive
preferences. Let (c(z t ), l(z t ), g(z t )) denote consumption, labor supply and
public good consumption of the representative household in period t after history z t . The household evaluates utility from a plan {(c(z t ), l(z t ), g(z t )), ∀t, z t }
7
using a recursive utility:
U (z t ) = W ((c(z t ), l(z t ), g(z t )), µ(U (z t+1 ))),
(2)
where W is an utility aggregator and µ ia a certainty equivalence function.
For more details see Backus et al. [2004]. In this paper we study a special
case with robust preferences:
Z
βb t
t
t
t
−θb U (z t+1 )
U (z ) = u(c(z ), l(z ), g(z )) − b ln
e
dF (zt+1 |zt ) .
(3)
θ
The period utility function u(c, l, g) is strictly increasing and strictly concave
in (c, −l, g). Parameter θb measures the degree of uncertainty aversion. But
it can also be interpreted as a risk-aversion to intra-temporal wealth gambles
as in Tallarini [2000]. We use the latter interpretation. This permits us to
calibrate θb using error-detection probabilities as suggested by Hansen and
Sargent [2007].
Households cannot borrow or lend and simply consume their own income in each period. Households take as given government’s current policy
(τ, g, b0 ) that consists of labor income tax rate τ , public good provision g and
borrowing b0 . Household’s problem then is:
max u(c, l, g) : c = (1 − τ )zl.
c,l
(4)
The optimality condition that we are going to exploit is:
−ul (c, l, g)/uc (c, l, g) = (1 − τ )z.
(5)
Together with the budget constraint it allows us solving for the optimal labor
supply policy l∗ (τ, z) given the tax rate t, the level of government spending
g, and the productivity level z. We denote the indirect utility function of the
household by w(τ, g, z):
w(τ, g, z) ≡ u(c∗ (τ, g, z), l∗ (τ, g, z), g),
(6)
where (c∗ , l∗ ) denotes the optimal choice of the household.
Finally, we can define the tax revenue function:
T (τ, g, z) ≡ τ zl∗ (τ, g, z).
(7)
It depends on the government spending to the degree that the optimal labor
supply does.
8
3.3
Government
Penalty for defaulting is a permanent or a temporary exclusion from the
international capital markets. When the government is in financial autarky
it cannot borrow or lend. It chooses tax and spending policies (τ, g) to
maximize the household’s welfare:
Z
βb t
t
t
−θb A(z t+1 )
A(z ) = max
[w(τ (z ), g(z ), zt )] − b ln
e
dF (zt+1 |zt ) (8a)
τ (z t ),g(z t )
θ
subject to a budget constraint:
g(z t ) 6 T (τ (z t ), g(z t ), zt ).
(8b)
We also define a value function Ad for the country that is suffers economic
penalties in addition to being banned from the international credit markets.
Penalty comes in the form of reduced productivity d(zt ) 6 zt during the default period. Value function Ad must satisfy the following Bellman equation:
Z
βb −θb A(z t+1 )
t
t
t
e
dF (zt+1 |zt )
Ad (z ) = max
[w(τ (z ), g(z ), d(zt ))] − b ln
τ (z t ),g(z t )
θ
(9)
With access to financial markets the government chooses tax, spending and
borrowing policies (τ, g, b0 ) to maximize household’s welfare:
n
V (z t ) = t max
w(τ (z t ), g(z t ), zt )
τ (z ),g(z t ),b0 (z t )
Z
o
βb b
t+1
t+1
e−θ max[V (z ,A(z )] dF (zt+1 |zt ) . (10a)
− b ln
θ
subject to a budget constraint:
g(z t ) + b(z t−1 ) 6 T (τ (z t ), g(z t ), zt ) + qb (b0 (z t ), zt )b0 (z t ).
3.4
(10b)
International creditors
International creditors are risk-neutral, two-period agents that rank alternative consumption streams according to:
Z
βc c
t+1
c t
t
U (z ) = c(z ) − c ln
e−θ c(z dF (zt+1 |zt ) .
(11)
θ
9
and discount future profits at gross rate R = 1/β c . Each creditor can supply
one unit of funds to the country-borrower.4
Finally, we assume that creditors are more patient than the household in
the country-borrower:
β c > β b.
(A)
3.5
Recursive representation and equilibrium
We restrict our attention to Markov perfect equilibria. We postulate that
government policies and repayment decision are functions of the current debt
b(z t ) and current productivity level zt only. Let A(z) be the autarkic welfare when current productivity is z. It must satisfy the following Bellman
equation:
Z
n
o
βb
−θb A(z 0 )
0
A(z) = max
w(τ,
g,
z)
−
e
dF
(z
|z)
.
(12)
g6τ zl∗ (τ,g,z)
θb
Let V (b, z) be the optimal life-time utility of the household when the government chooses to repay its debt b and productivity is z. It must satisfy the
following Bellman equation:
Z
βb
−θb max[V (b0 ,z 0 ),A(z 0 )]
0
V (b, z) =
max
w(τ, g, z) − b
e
dF (z |z)
(τ,g,b0 )∈B(b,z)
θ
(13a)
where the budget set B(b, z) is defined as follows:
n
o
B(b, z) = (τ, g, b0 ) : g + b 6 T (τ, g, z) + qb (b, z)b0 .
(13b)
Let 1r (b, z, z 0 ) ≡ 1(Ad (z 0 ) 6 V (b0∗ (b, z), z 0 )) denote a debt repayment indicator. Then creditor’s individual rationality implies:
qb (b, z) = βc δ̃(b, z),
where
Z
δ̃(b, z) =
1r (b, z, z 0 )dF̃ (z 0 |z)
(14)
(15)
z
4
This assumption implies that the bond price is not a function of the amount a single
creditor lends but only the gross debt outstanding.
10
is the next period’s ‘twisted’ probability of repayment. It is computed with
respect to the ‘twisted’ probability measure F̃ :
c
0
e−θ 1r (b,z,z ) dF (z 0 |z)
.
dF̃ (z |z) ≡ R −θc 1r (b,z,z0 )
e
dF (z 0 |z)
0
(16)
Given the above we can write:
δ̃(b, z) =
e−θc δ(b, z)
,
e−θc δ(b, z) + 1 − δ(b, z)
(17)
where δ(b, z) is the true, i.e. computed using F , probability of repayment.
Definition. A recursive competitive equilibrium is debt price function qb , government’s tax, spending and borrowing policies (τ, g, b0 ) and household’s consumption and labor supply policies c, l such that:
1. (c, l) solve the household’s optimization problem (4);
2. (τ, g, b0 ) solve the government’s optimization problem (13a-13b);
3. qb satisfies the creditor’s individual rationality condition (14).
3.6
Simplified formulation
The first-order optimality condition of the government is:
∂
∂
wτ (τ, g, z)
T (τ, g, z) − 1 = wg (τ, g, z) T (τ, g, z).
∂g
∂τ
(18)
The above equation implicitly defines the optimal public spending g ∗ (t, z).5
Using the optimal spending policy the optimization problem of the government can be reformulated in terms of (t, b0 ) alone. So, we can rewrite (13a)
as follows:
Z
∗
b
0 0
0
V (b, z) = max
w(τ, g (τ, z), z) + β
V (b , z )dF (z |z)
(19)
0
(t,b )
z0
subject to the “indirect” budget constraint:
g ∗ (τ, z) + b = T (t, g ∗ (t, z), z) + qb (b, z)b0 .
5
When the utility function is separable in the government’s spending as it will be
assumed then (18) uniquely defines g ∗ .
11
For any choice of states (b, z) this is effectively a one-dimensional optimization problem that can be solved using standard numerical techniques. Let
τ ∗ (b, z) and b0∗ (b, z) denote respectively the optimal tax and debt policies of
the government.
We conclude this section with a useful proposition. It shows that the
government tax revenue, spending, and budget surplus are strictly monotone
functions of t for all feasible tax rates. That is there is a unique way of
implementing any desired level of government surplus. The implication of
this result is that for any choice of (b, z) there is a unique solution to the
government’s optimization problem.6
Proposition 1. Let z > 0. The following statements are true on the tax
rate interval [0, (1 + η)−1 ]:
A. The government tax revenue T (t, z) is a str. increasing function of t;
B. The government spending g(t, z) is a str. decreasing function of t;
C. The government budget balance S(t, z) is a str. increasing function of t.
Proof. This can be shown directly by differentiation using formulas in the
appendix A.
Finally, because S(t, z) is a strictly monotone function of t and S(0, z) <
0 < S((1 + η)−1 , z), ∀z there exists unique taut (z) ∈ (0, (1 + η)−1 ), ∀z such
that S(taut (z), z) = 0. The latter is the optimal level of taxation in autarky.
Figure 5 plots the government budget surplus and the government spending for several choices of η: 0.5, 1.0, and 2.0. Panel A demonstrates the result
stated in proposition 1. Panel B shows that the government spending is a
decreasing function of the tax rate t. While the government tax revenue increases with the tax rate most of this revenue is saved by the government. It
is optimal to do so to keep the desired balance between the declining private
consumption and the public good consumption.
6
This also aids computations as the relation between government deficit and tax rate
can be pre-computed.
12
A. Budget balance (surplus), S(τ,z)
B. Government spending, g(t,z)
0.40
0.50
0.30
0.45
0.20
0.40
0.35
0.10
0.30
0.00
0.25
-0.10
0.20
-0.20
0.15
-0.30
0.10
η=0.5
η=1.0
η=2.0
-0.40
0.05
-0.50
0.00
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0
tax rate, τ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
tax rate, τ
Figure 5: Government budget surplus and spending as functions of the tax
rate. Parameters: γ = 2, χ = 0.5.
4
Results
We illustrate our findings with a numerical example. Assume that the household’s one period utility is:
l1+1/η + χln(g).
(20)
u(c, l, g) = ln c − a
1 + 1/η
We assume that η = 1.00, a typical labor supply elasticity. We calibrate φ so
that government spending is 15% of the country’s GDP. Productivity process
is assumed to be an iid log-normally distributed random variable. The world
interest rate is 5%. We would like the reader to regard this parameterization
as an example rather than a rigorous calibration.7
We calibrate model mis-specification parameters to match the desired
error-detection probability as suggested by Hansen and Sargent [2007]. The
7
Notice that most of the moments – output volatility, level of government spending,
etc. – are endogenous and the model must be calibrated using the simulated method of
moments.
13
parameter
value moment
βb
0.915-0.943 default probability is 0.75%
βc
0.987 world interest rate is 6%
γ
2.000 ‘typical’ value
η
1.000 ‘typical’ value
χ
0.300 gov. spending is 13.4% of GDP
λ
0.282 Recovery probability
d
0.850 Average output loss after default is 1-2%
c
θ
1.200 creditor’s edp is ≈0.40
b
θ
0.800 borrower’s edp is ≈0.40
Table 1: Calibrated parameters
error-detection probability (EDP) is the probability that the likelihood ratio
model test errs. Let E1 denote occurrence of the type-I error – when a true
model is incorrectly rejected. Let E2 denote occurrence of type-II error –
when a wrong model is incorrectly accepted. Then
EDP = 0.5[prob(E1 ) + prob(E2 )] ∈ [0, 0.5].
(21)
Low EDP signifies that a researcher would err infrequently. This implies that
models are easy to distinguish. If Fe is a ‘twisted’ version of F then:
EDP = 0.5[prob(F accepted|Fe) + prob(Fe accepted|F )].
The above can be computed by Monte-Carlo simulation. Model π is accepted
when it generates higher likelihood of a simulated sample than its alternative
π̃. The larger the sample size the higher is EDP. We set sample size to 74,
the length of the Argentina’s data. We choose θc and θb such that EDP is
0.40.8 That is if a random sample of length 74 is generated an econometrician
would correctly discriminate the two models with probability of only 20%.
The model solution is computed using collocation method for the Bellman
equation. Collocation grid is a tensor product of 300 values of z and 500
values of b. The algorithm is stopped when a change in the implied interest
rate, 1/q(b), is less than one tenth of a basis point.
8
A typical value used is 0.25.
14
A. Consumption, c
B. Next period assets, b’
0.80
0.20
45o
0.70
0.10
0.60
0.00
0.50
0.40
-0.05
zmin
zmax
0.00
0.05
0.10
-0.10
-0.05
0.15
C. Government spending, g
0.00
0.05
0.10
0.15
D. Bond price, q
0.25
1.00
0.80
0.20
0.60
0.15
0.40
0.10
0.05
-0.05
0.20
0.00
0.05
0.10
0.00
-0.05
0.15
E. Tax rate, τ
0.00
0.05
0.10
0.15
0.10
0.15
F. Value fn, V
0.30
-70.0
-71.0
-72.0
0.25
-73.0
0.20
-74.0
-75.0
0.15
-76.0
-77.0
0.10
-0.05
0.00
0.05
0.10
-78.0
-0.05
0.15
0.00
0.05
Figure 6: Model solution when both creditors and the borrower are concerned
about model mis-specification: θc = ln(1.2), θb = 0.8. Government’s assets,
a, are measured on the x -axis. Each line corresponds to z = µz + iσz , i =
−3, −2, ..., 3.
Figure 6 demonstrates the model solution. The lowest asset level observed
in equilibrium is -0.018. Panel A plots the household’s consumption level. It
15
is increasing in the productivity level z. But it is not a monotone function
of the government’s assets b. But the region where non-monotonicity is
observed is never reached in equilibrium. So, consumption is a monotone
function of the government’s assets and the productivity level. The same is
true about the public spending depicted in panel C. Both public and private
consumption are driven by the tax rate that decreases monotonically as the
government’s assets increase. When the government is in debt the tax rate
varies little with the productivity level and it stays close to its maximum level
of 19%. That is the government chooses to tax heavily to limit the decline
in government spending. When the government has a sufficient amount of
assets it taxes labor less, especially when the productivity level is low.
Panel D plots the bond price function q(b). It is a monotone function
of assets and largely reflects the shape of the cdf of the productivity shock
z. While the bond price can potentially reach zero this does not happen in
equilibrium as the government, “warned” with a ballooning borrowing cost,
increases taxes sharply. Panel F plots the optimal value for the government
when it decides to repay (increasing black lines) and when it decides to default
(horizontal gray). The circles and the line that connects them marks the
“default boundary.” These are points at which the government is indifferent
between repaying and defaulting.
Public spending declines slowly (note the scale) as government’s assets
decline. That is the government chooses to tax labor more rather than to
lower spending. In low productivity states such policy depresses output further and can lead to a crisis. Finally panel E plots an equilibrium bond price
function. It largely reflects the distribution of the exogenous productivity
state z.
We now turn to the “twisted” distributions. The government-borrower
in this setting behaves as if having ‘twisted’ beliefs. Its beliefs are determined endogenously in equilibrium as the ‘degree’ of twisting depends on
the equilibrium outcomes:
e−θb max[V (b,z),Ad (z)]
.
e−θb max[V (b,z),Ad (z)] dF (z)
z
dFeb (z|b) = R
(22)
Figure 7 plots ratios of distributions used by creditors and the borrower. Both
16
relative likelihood, dFtilde(z)/dF(z)
1.20
borrower
creditor
1.15
1.10
1.05
1.00
0.95
0.90
0.85
0.80
0.90
0.95
1.00
1.05
1.10
productivity, z
Figure 7: Ratios F̃ (z)/F (z) for θc = 1.20, θb = 0.80
twist their distributions pessimistically. For creditors adverse outcomes are
those that lead to default. That is why creditors overestimate probability of
outcomes in the left tail: F̃ c (z)/F (z) > 1 for z 6 0.961. The borrower twists
the whole distribution. So, F̃ b (z)/F (z) > 1 for below the mean productivity
levels, z 6 1, and F̃ b (z)/F (z) < 1 for above the mean productivity levels,
z < 1. If we consider F̃ c (z)/F̃ b (z) there are three regions. This ratio is above
one at the lower tail and the upper body of the distribution. This ratio is
below one at the lower part of the distribution body.9 Disagreement leads
to volatile returns because as the borrower’s wealth changes market power
shifts between the borrower and creditors reflecting respective beliefs. With
multiple assets endogenous disagreement would lead to frequent changes of
market volume in select markets as in Routledge and Zin [2009]. But in our
model we only have one asset; so, volatile demand driven by the disagreement
must be reflected in more volatile prices and returns.
Figure 8 demonstrates the economy’s dynamics along the path of low
productivity realizations. The initial debt is assumed to be zero. As can be
seen from the figure the debt is steadily increasing. The sovereign lending
premium, that equals the “twisted” default probability here, stays zero until
9
See also figure 10 in the appendix for “twisted” distributions.
17
A. Debt
B. Premium
0.050
0.030
0.025
0.000
0.020
-0.050
0.015
-0.100
0.010
-0.150
0.005
-0.200
0.000
0
2
4
6
8
10
0
C. Tax rate
2
4
6
8
10
8
10
D. Public spending
0.050
0.046
0.045
0.045
0.040
0.035
0.044
0.030
0.043
0.025
0.020
0.042
0.015
0.041
0.010
0.040
0.005
0.000
0.039
0
2
4
6
8
10
0
2
4
6
Figure 8: A run-up to a crisis: a sequence of shocks is {e−σz , e−σz , ...}
period 5 when it jumps first above 0.01 then to 0.03. The tax rate is following
the same path. An increased tax rate is needed to generate a larger surplus
and so to slow-down debt accumulating. Not to discourage production the
government balances higher taxes with lower public good provision. Betting
on better productivity levels to realize in the near future the government
is slowly led into default “zone.” While default does not occur along the
assumed path of productivity shocks there is a significant probability of nonpayment starting with period t=5.
Figure 9 presents simulated series. Default episodes correspond to periods
where premium is negative. Panel A shows that government debt is very
volatile. The government spends about a third of the time with no debt.
18
But government savings are decumulated quickly. So, a crisis-like situation
can arise quickly. Bond returns are also volatile with perceived probability of
default often rising above 4%. Despite this most of the time bond premium
is zero. In the data the premium is always above zero even at times when
the government’s debt is relatively low. One possibility to fix this could be
to introduce disaster shocks like in Adam and Grill [2011] or a possibility of
partial repayment.
A. Government debt
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
50
100
150
200
250
300
350
400
450
500
350
400
450
500
B. Premium
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
50
100
150
200
250
300
Figure 9: Model simulation. Periods were the premium is negative are default
periods.
Table presents model moments. It contains four columns. The first column is the baseline parameterization in which neither the borrower nor creditors have concerns for model mis-specification. The second and the third
columns turns on model misspecification for creditors and the borrower (one
at a time) respectively. The fourth column features two-sided model mis19
specification. We vary the discount factor of the borrower to keep the default
probably close to constant.
We start with bond premia that is the focus of this paper. The mean
bond premium does not vary much across specifications and is the range of
3.39 − 4.31%. That much lower than in the data but significant given how
small is βc − βb relative to other studies. On the other hand, volatility of
bond premium varies substantially across different specifications. When we
turn on model mis-specification for creditors bond premium volatility goes
up only marginally: from 6.51% to 7.12%. But when the borrower also has
concerns about model misspecification then it increases to 8.21%. Interestingly that the two forms of model misspecification reinforce each other.
This occurs for the following reason. If creditors have concerns for model
misspecification country premium increases but the country borrower then
contracts its borrowing. So, while the interest rate schedule becomes steeper
high interest rates are not observed in equilibrium. Now suppose that the
country-borrower also has concerns for model misspecification. Its expectations shift in the direction towards more pessimism. The value of staying
in the market declines but the value of defaulting declines even more. So,
the government willingly undertakes high interest loans because of fears of
autarky. Thus, it is a robust borrower’s fear that leads him to borrow more,
at higher rates, and not default longer than optimally.
We will now discuss fiscal policy. As we turn on additional levels of
model misspecification taxes become smaller and more volatile. That is as we
introduce more pessimistic into the model we increase precautionary motives
for the government. It taxes less on average and it also taxes more heavily
in good times. That is fiscal policy becomes more “counter-cyclical.” The
benefit of doing so is that the government can be more accommodating when
productivity is low. This can be seen from the correlation ρz<µz (τ, z) that
conditions on productivity being below its average. The government controls
the level of pessimism present in equilibrium by not taxing the economy
heavily in adverse states of the world.10 In other words, the government is
managing expectations as in Karantounias [2013].
10
Taxes still increase but not as much as they otherwise would.
20
θb = 0.00 θb = 0.00 θb = 0.80 θb = 0.80
θc = 0.00 θc = 1.20 θc = 0.00 θc = 1.20
βb = .943 βb = .938 βb = .938 βb = .915
µ[Rb ]
3.66
4.11
3.39
4.31
σ[Rb ]
6.51
7.12
7.95
8.21
ρ[Rb , y]
-.71
-.69
-.75
-.75
µ(y)
0.82
0.82
0.83
0.82
σ(y)
7.29
7.30
7.29
7.29
µ(τ )
0.22
0.33
0.12
0.11
σ(τ )
0.40
0.48
0.62
0.65
ρ(τ, z)
0.41
0.28
0.81
0.89
ρz<µz (τ, z)
0.34
0.23
0.48
0.49
1 − µ(pr )
0.75
0.75
0.75
0.75
b
EDP
0.50
0.50
0.38
0.38
c
EDP
0.50
0.41
0.50
0.41
Table 2: Model moments
5
Conclusions
This paper builds a model of sovereign debt and default with an endogenously
determined fiscal policy and two-sided concerns of model misspecification.
Allowing for an endogenous fiscal policy enabled us to talk about fiscal deficit
and government debt in pre-crisis periods. We showed that, like in the data,
government deficit shrinks and even turns into a surplus, driving the current
account out of red. Allowing for two-sided concern for model misspecification
enabled us to improve the model’s predictions about sovereign bond spreads.
While we enriched the model substantially computational complexity remains almost as that of simple exchange economies. This opens a possibility
for modelling such first-order-of-importance factors as capital accumulation,
political risk, and monetary policy.
21
References
Klaus Adam and Michael Grill. Optimal sovereign debt default. Mannheim
University manuscript, 2011.
Mark Aguiar and Manuel Amador. Growth in the shadow of expropriation.
Quarterly Journal of Economics, 126(2):651697, 2011.
Mark Aguiar and Gita Gopinath. Defaultable debt, interest rates and the
current account. Journal of International Economics, 69(1):64–83, 2006.
S. Rao Aiyagari, Albert Marcet, Thomas J. Sargent, and Juha Seppala. Optimal taxation without state-contingent debt. Journal of Political Economy,
110:1220–1254, 2002.
Cristina Arellano. Default risk and income fluctuations in emerging
economies. The American Economic Review, 98(3):690–712, 2008.
David K. Backus, Bryan R. Routledge, and Stanley E. Zin. Exotic Preferences
for Macroeconomics, volume 19, pages 319–414. MIT Press, 2004.
Andrea Ferrero. A structural decomposition of the U.S. trade balance: Productivity, demographics and fiscal policy. Journal of Monetary Economics,
57(4):478–490, 2010.
Lars P. Hansen and Thomas J. Sargent. Robustness. Princeton University
Press, 2007.
Anastasios Karantounias. Managing pessimistic expectations and fiscal policy. Theoretical Economics, 8:193–231, 2013.
Demian Pouzo. Optimal taxation with endogenous default under incomplete
markets. UC Berkeley manuscript, 2013.
Bryan R. Routledge and Stanley E. Zin. Model uncertainty and liquidity.
Review of Economic Dynamics, 12:543–566, 2009.
Thomas D. Tallarini. Risk-sensitive real business cycles. Journal of Monetary
Economics, 45(3):507–532, 2000.
and many more to be cited later...
22
A
Derivations for CRRA-GHH preferences
Assume the following utility specification:
u(c, l, g) =
1 h
l1+1/η i1−γ
g 1−γ
c−a
.
+χ
1−γ
1 + 1/η
1−γ
(23)
The solution to the household optimization problem is the labor supply function l∗ (t, z):
l∗ (t, z) = ((1 − t)z/a)η .
(24)
The indirect (period) utility function of the household then is:
a−(1−γ)η ((1 − t)z)(1−γ)(1+η)
g 1−γ
u((1 − t)l (t, z), l (t, z), g) =
+χ
.
(1 − γ)(1 + η)
1−γ
∗
∗
(25)
The government’s FOCs allow to express the government spending g as a
function of the tax rate t:
g ∗ (t, z) = a−η
z 1+η 1/γ
χ (1 − t)1+η−1/γ (1 − (1 + η)t)1/γ .
1+η
(26)
Because government spending must be positive the above equation implies
1
that the tax rate cannot exceed 1+η
. This bound decreases as the elasticity
of labor supply η increases.
Because the government spending and the labor supply are functions of
(t, z) so is the government tax revenue, T (t, z), and primal surplus, S(t, z):
T (t, z) ≡ tzl(t, z) = a−η z 1+η (1 − t)η−1 [1 − (1 + η)t)]
S(t, z) ≡ tzl(t, z) − g(t, z)
= a−η
1+η
z
1+η
(27)
(28)
h
i
(1 − t)η (1 + η)t − [χ(1 − t)γ−1 (1 − (1 + η)t)]1/γ .
These results are used in the proof of proposition 1.
23
B
Derivations for the special case with γ = 1
With preferences specified in (20) we get the following relations:
l∗ (τ, g, z) = [(1 − τ )z/a]η ,
−η
w(τ, g, z) = ln(a [(1 − τ )z]
−η 1+η
T (τ, g, z) = a z
(29a)
1+η
/(1 + η)) + φln(g),
η
τ (1 − τ ) .
(29b)
(29c)
We now derive the optimal spending - tax relation:
g ∗ (τ, z) = φa−η z 1+η (1 − τ − ητ )(1 − τ )η .
(30)
It is easy to show that g(τ, z) is positive and strictly decreasing function
for all τ 6 1/(1 + η). That is there is a monotone relation between the
desired spending and taxation levels. The total tax revenue is also a strictly
increasing function of tax rate when τ 6 1/(1 + η).
Fiscal deficit is:
d(τ, z) ≡ g ∗ (τ, z) − T (τ, g ∗ (τ, z), z)
= a−η z 1+η (1 − τ )η [φ − τ (1 + φ + φη)].
(31)
It is a decreasing function of tax rate when τ 6 1/(1 + η). That is for
any level of fiscal deficit there is a unique tax level that “implements” it.
With access to financial markets the government chooses (τ, b0 ) subject to
d(τ, z) = qb (b0 , z)b0 (b, z) − b. In financial autarky the optimal tax rate is
determined from d(τ, z) = 0 as b0 = b = 0:
τA∗ (z) =
φ
,
1 + φ + φη
24
∀z.
(32)
0.05
true
borrower
creditor
0.04
density dF(z)
0.04
0.04
0.03
0.03
0.02
0.01
0.01
0.01
0.00
0.90
0.95
1.00
1.05
1.10
productivity, z
Figure 10: F̃ c (z), F̃ b (z) and F̃ c (z) for θc = 1.20, θb = 0.80
25