where ηy is the semi-elasticity of the government budget (relative to GDP) with respect to
output, ηπ is the semi- elasticity of the budget with respect to the price level and Et denotes
expectations conditional on the information at time t.
Hence, the surplus-to-GDP ratio may be higher than expected because of unanticipated
output, yt+1 - Etyt+1, and inflation, πt+1 - Etπt+1. While the impact of economic activity
on the budget is well known from a number of studies, inflation also reduces the deficit if
tax systems and spending programs are not fully indexed.5
3. The choice of debt denomination and indexation
The objective of the Treasury is to minimize the probability that debt stabilization
fails because the adjustment effort is unsuccessful; because revenues falls short of expected
and/or spending programs cannot be cut. The government chooses s, q and h to minimize
Prob[X > At+1 - ∆BtT+1]=Min Et ∞ φ(X)dx
At+1 -∆BtT+1
Min Et
(5)
subject to (2), (3) and (4).
where φ(X) denotes the probability density function of X.
Deriving (5) with respect to s, q and h yields
Etφ(At+1 - ∆BtT+1)[it+1 - Rt]=0 (6)
Etφ(At+1 - ∆BtT+1)[RtUS + RPt + et+1 - et - Rt]=0 (7)
Etφ(At+1 - ∆BtT+1)[RtI + πt+1 - Rt]=0 (8)
where At+1 -∆BtT+1 is the planned reduction in the debt-to-GDP ratio and φ(At+1 -∆BtT+1)
is a function of s, q and h.
The first order conditions (6)-(8) have a simple interpretation: they shows that the
debt structure is optimal only if the increase in the probability of failure that is associated
with the interest cost of additional funding, in a particular type of debt, is equalized across
debt instruments. If this were not the case, the government could reduce the probability of
failure by modifying the debt structure; i.e. it could substitute fixed-rate bonds for Selic-rate
indexed bonds or vice versa.6
To gain further intuition we observe that the difference between the interest cost of
Selic-indexed bonds and fixed-rate bonds is equal to the difference between the (average)
Selic rate between time t and t + 1 and its value as expected at the time t, minus the term
premium on fixed-rate bonds:
(9)
it+1 - Rt = it+1 - Etit+1 - TPt
5in output by one percent are reported in the OECD Economic Outlook (1999) and Van der Noord (2000).
With the notable exceptions of Austria, Denmark, Ireland, the Netherlands and Sweden, OECD countries
have elasticities in the 0.4 to 0.7 range. The effects of inflation on government budgets have not been
measured to the same extent, but appear substantial. For Sweden, Persson, Persson and Svensson (1998)
estimate a budget improvement of 0.4% of GDP on a yearly basis for a one percent increase in the inflation
rate.
6 The argument assumes that there are non-negative constraints to the choice of debt instruments.