Theoretical prerequisites for fiscal sustainability analysis, C Burnside

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Content: Theoretical Prerequisites for Fiscal Sustainability Analysis Craig Burnside July 2003 To deЮne Юscal sustainability analysis it is useful to seek guidance from a dictionary. Webster's, for example, suggests using the adjective sustainable to describe something that can be kept up, prolonged, borne, etc. Or, it might be used to describe a method of harvesting a resource so that the resource is not depleted or permanently damaged in the process. When we speak of Юscal sustainability we are typically referring to the Юscal policies of a government. Of course, we must make our deЮnition of Юscal sustainability more precise than the dictionary deЮnitions given above. On the other hand, they can guide our thinking. The resource depletion analogy is not entirely appropriate, because a government's resources are not that comparable to mineral, or other, resources. On the other hand, it does suggest a concept of sustainability related to solvency. When we speak of solvency we refer to the government's ability to service its debt obligations without explicitly defaulting on them. One concept of Юscal sustainability relates to the government's ability to indeЮnitely maintain the same set of policies while remaining solvent. If a particular combination of Юscal and/or monetary policy would, if indeЮnitely maintained, lead to insolvency, then we refer to these policies as unsustainable. One role of Юscal sustainability analysis is to provide some indication as to whether a particular policy mix is sustainable or not. Often governments will change their policies if it becomes clear that they are unsustainable. Thus, the focus of Юscal sustainability analysis is frequently not on default itself­which governments frequently avoid­rather it is on the consequences of the policy changes needed to avoid eventual default. University of Virginia
Even when a government is solvent, and is likely to remain solvent, its Юscal policies may be costly. Sometimes Юscal sustainability analysis will refer to the ongoing costs associated with a particular combination of Юscal and monetary policies. In the rest of this chapter we develop the simple theoretical framework within which Юscal sustainability analysis is usually conducted. We will introduce several concepts: the singleperiod government budget constraint, the lifetime budget constraint, the Юscal theory of the price level, the no-Ponzi scheme condition, and the transversality condition. In Chapter 2 we will show how these tools can be used in analyzing and Interpreting data.
1. The Government Budget Constraint
The fundamental building block of Юscal sustainability analysis is the public sector or government budget constraint, which is an identity:
net issuance of debt = interest payments - primary balance - seigniorage. (1.1)
The net issuance of debt is gross receipts from issuing new debt minus any amortization payments made in the period.1 The identity, (1.1), can be expressed in mathematical notation as
Bt - Bt-1 = It - Xt - (Mt - Mt-1).
(1.2)
Here the subscript t indexes time, which we will usually measure in years, Bt is quantity of public debt at the end of period t, It is interest payments, Xt is the primary balance (revenue minus noninterest expenditure), and Mt is the monetary base at the end of period t, all measured in units of local currency (LCUs). The only subtlety involved in (1.2) is in associating the net issuance of debt, a net cash Яow, with the change in a stock, the quantity of debt, Bt. To be sure that these objects are equivalent, we should be more precise as to the deЮnitions of the quantity of debt and interest payments. To deЮne the former, we must decide how to value the government's outstanding debt obligations. To deЮne the former we must divide debt service between amortization and interest. We will not explore precise deЮnitions of debt and interest until Chapter 2. Two clarifying statements should be made at this point. First, both the debt and interest payments concepts should be net; i.e. debt should be net of any marketable assets and 1The government might also issue new debt in order to Юnance the repurchase of old debt. In this case, we would still be concerned with the net proceeds raised.
2
interest should be payments net of receipts. Second, the analysis must Юx on a particular deЮnition of the government or public sector since different measures of the variables in (1.2) would apply to different deЮnitions of the public sector. For example, by including seigniorage revenue (the change in the monetary base) in our deЮnition, we have implicitly deЮned the public sector to include, at least, the Central Bank in addition to the central government. It is probably most conventional to deЮne the public sector as the consolidation of the central government, state and local governments, state-owned nonЮnancial enterprises and the central bank. Sometimes state banks are also included in the deЮnition of the public sector, but more commonly they are not. As we will see in Chapter 2, (1.2) is the fundamental building block for studying the evolution of the government's debt over time, or, using a common phrase, the government's debt dynamics. But the Яow budget constraint is also the Юrst step in deriving the lifetime government budget constraint, which plays a crucial role in assessing a government's Юnances, interpreting its Юscal policies, and predicting the consequences of particular shocks to the economy for prices and exchange rates. To derive the lifetime budget constraint we need to Юrst rewrite the Яow budget constraint. To begin, we will assume that time is discrete, that all debt has a maturity of one period, that debt is real, in the sense that its face value is indexed to the price level, and pays a constant real rate of interest, r. In this case (1.2) can be rewritten as
bt = (1 + r)bt-1 - xt - t,
(1.3)
where bt = Bt/Pt is the end-of-period t stock of real debt, xt = Xt/Pt is the real primary surplus and t = (Mt - Mt-1)/Pt is the real value of seigniorage revenue.2 Rearranging (1.3) we have
bt-1 = (1 + r)-1bt + (1 + r)-1(xt + t).
(1.4)
Notice that the expression for bt-1, on the right-hand side of (1.4), can be updated to period
t, implying that
bt = (1 + r)-1bt+1 + (1 + r)-1(xt+1 + t+1).
(1.5)
2The correspondence between (1.2) and (1.3) can be veriЮed if one assumes that the interest payment includes an indexation adjustment for both the interest and the principal on the loan. I.e.:
It = [(Pt/Pt-1)(1 + r) - 1]Bt-1.
3
This can be used to substitute for bt on the right-hand side of (1.4): bt-1 = (1 + r)-2bt+1 + (1 + r)-1(xt + t) + (1 + r)-2(xt+1 + t+1).
(1.6)
Clearly, the same procedure could be used to substitute for bt+1 on the right-hand side of
(1.6), and then for bt+2, etc., in a recursive fashion. Hence, after several iterations we would
obtain
bt-1
=
(1
+
r)-(j+1)bt+j
+
!j (1
+
r)-(i+1)(xt+i
+
t+i).
i=0
(1.7)
Equation (1.7) provides a link between the amounts of debt the government has at two
dates: t - 1 and t + j. In particular, the amount of debt the government has on date t + j is
a function of the debt it initially had at date t - 1, as well as the primary surpluses it ran,
and seigniorage it raised between these dates.
If we impose the condition
lim (1 j
+
r)-(j+1)bt+j
=
0
(1.8)
then we obtain what is frequently called the government's lifetime budget constraint:
bt-1
=
! (1
+
r)-(i+1)(xt+i
+
t+i).
i=0
(1.9)
Intuitively, the lifetime budget constraint states that the government Юnances its initial debt
by raising seigniorage revenue and running primary surpluses in the future, whose present value is equal to its initial debt holdings.3
The lifetime budget constraint is a fundamental building block for a number of different
tools and theoretical arguments developed in the literature and discussed here. In the next
section we use the lifetime budget constraint as a theoretical tool to discuss the effects of
government deЮcits on inЯation. This discussion will be closely related to the theoretical
arguments made in Sargent and Wallace's (1981) unpleasant monetarist arithmetic paper
and Sargent's (1983, 1985) papers on monetary and Юscal policy coordination. Later we will
show that in the Юscal theory of the price level (1.9) is reinterpreted as an equation that prices government debt.4
3In deriving (1.9) we imposed the condition (1.8) without stating its origin. Absent a theoretical model, of course, there is no reason for imposing this condition, as it is not obvious, without a theory, why the stock of debt should evolve in a way that satisЮes (1.8). The end of this chapter discusses the appropriate interpretation of (1.8). 4See Sims (1994), Woodford (1995) and Cochrane (2001a, b), which discuss the Юscal theory in a closed economy context. Dupor (2000), Daniel (2001), and Corsetti and Mackowiak (2002) analyze the implications of the Юscal theory for open economies.
4
In Chapter 2 we use the lifetime budget constraint to derive our a simple tool for assessing Юscal sustainability: the long-run sustainability condition. We will also show how it can be used as the basis of formal statistical tests of (1.8), as in Hamilton and Flavin (1986).
2. Fiscal and Monetary Policy and the Effects of Government DeЮcits
In this section we will discuss the effects of government deЮcits on inЯation. We will also discuss the issue of Юscal and monetary policy coordination. At Юrst, it may not seem obvious that these questions and concerns are closely related to Юscal sustainability. In fact, they are intimately related to Юscal sustainability. As was argued in the introduction, it is possible that a government will violate (1.9) by defaulting on its debt obligations. However, in many cases, the role of Юscal sustainability analysis is not to point out concerns about default. Rather, the role of the analysis is to discuss the macroeconomic consequences of alternative policies which happen to all be consistent with (1.9). The lifetime budget constraint can be satisЮed by generating large primary surpluses, but it can also be satisЮed by generating a lot of seigniorage revenue. Obviously these policies will have different consequences for macroeconomic outcomes, and the timing of when surpluses occur may also be important. This section take a Юrst step in the direction of understanding the impact of different policy choices. Much of the discussion is closely related to the arguments outlined in Sargent and Wallace (1981), and Sargent (1983, 1985). We begin by returning to the real version of the government budget constraint, (1.3):
bt = (1 + r)bt-1 - xt - (Mt - Mt-1)/Pt.
Notice that we can rearrange this equation as
Pt(bt - bt-1) + Mt - Mt-1 = Pt(rbt-1 - xt).
(2.1)
The right-hand side of (2.1) is the government's Юnancing requirement: the nominal value of real interest payments plus the primary deЮcit. The left-hand side of (2.1) is the government's Юnancing, which is a mix of net issuance of debt, and net issuance of base money. If one thinks of a Юscal authority (a legislature together with the ministry of Юnance) as the determiner of xt then, given that rbt-1 is pre-determined, (2.1) appears to deЮne the role of the monetary authority as essentially a debt manager: it picks the Юnancing mix between debt and monetary obligations.
5
Price Level Determination Conditional on a value for the price level Pt, we might think of the Юscal authority choosing xt, and then the monetary authority choosing Mt and bt consistent with (2.1). Not surprisingly, however, the price level, itself, will be inЯuenced by the monetary authority's choice. The link between the monetary authority's decision
making and the price level is usually made by writing down a model of money demand.
There are several models of money demand that we might use. For example, we might
use the quantity theory of money, whereby the demand for real money balances is simply,
Mt/Pt = yt/v, where v represents a constant value for the velocity of money and yt represents real GDP. Alternatively we might use a variant of the Cagan money demand representation in
which the demand for real balances depends negatively on the nominal interest rate: Mt/Pt = Ayt exp(-Rt), where Rt represents the nominal interest rate. Alternatively, we could use any simple model of money demand consistent with the basic assumptions in a standard
intermediate macroeconomics textbook, or perhaps a complicated general equilibrium model.
For the remainder of this chapter we will use either the quantity theory or a variant of Cagan
money demand, but mainly for analytical convenience. The qualitative Юndings we get with
these assumptions would be robust to other speciЮcations.
To obtain our Юrst results on the effects of policy on the price level we take the Cagan
money demand speciЮcation described above assuming that (i) the transactions motive is
constant, i.e. yt = y for all t, and (ii) the nominal interest rate is just Rt = r+Et ln(Pt+1/Pt). In this case we have
ln(Mt/Pt) = a - Et ln(Pt+1/Pt)
(2.2)
where a = ln(Ay) - r.
Notice that (2.2) represents a linear Юrst order difference equation in ln Pt. If we treat Mt as an exogenous stochastic process controlled by the central bank, (2.2) implies the following
solution for ln Pt:
1 ! " #j ln Pt = -a + 1 + j=0 1 + Et ln Mt+j
(2.3)
Importantly, Pt depends on the current money supply as well as the expected path of the
money supply. If the interest semi-elasticity of money demand, , is very small, say ap-
proximately 0, then the price level depends mainly on the current money supply: ln Pt -a + ln Mt. On the other hand, if is very large the discount factor in (2.3) will be close to 1, and the price level will depend a lot on what agents expect the money supply to be well
6
into the future.
Are Government DeЮcits InЯationary? In this section we use the solution for the price level, (2.3), to ask whether government deЮcits are inЯationary. We will see that this depends on how monetary and Юscal policy, together, are conducted. We will consider different policy regimes which have very different implications for inЯation.
Regime 1 Suppose the government follows a regime in which it issues debt to Юnance deЮcits, and money is never printed:
Mt = M for all t.
(2.4)
Under this regime, the lifetime budget constraint, (1.9), implies
b-1
=
! (1
+
r)-(t+1)xt
t=0
(2.5)
so that the present value of the primary balance is the initial debt stock. In essence, if we abstract from the stock of initial debt, the monetary policy Mt = M for all t implies that running a primary deЮcit at time 0, x0 > 0, forces there to be future primary surpluses in present value terms: i.e. $ t=1(1 + r)-(t+1)xt > 0. In this policy regime monetary policy is rigid, and future Юscal policy must tighten if current Юscal policy becomes looser. Furthermore, and as a consequence of this, notice that inЯation is zero in this regime: Pt = e-aM for all t. So running a deЮcit at time 0 causes no inЯation at time 0. This is precisely because agents in the economy understand the nature of monetary policy. They know that the deЮcit at time 0 will not be monetized at time 0, nor at any time in the future.
Regime 2 In the previous example there is no connection between the current primary
deЮcit and the inЯation rate. Now imagine a policy regime in which the government never
issues debt:
bt = 0 for all t.
(2.6)
Of course, in this setting the Яow budget constraint, (1.3), implies
Mt - Mt-1 = -Ptxt.
(2.7)
7
Not surprisingly, this policy regime is much more likely to be one where there is a connection between the current deЮcit and current inЯation. For example, in the extreme case where the interest rate elasticity is zero ( = 0), (2.3) and (2.7) imply Pt = e-aMt-1/(1 + e-axt). This means that the smaller the primary surplus is, the higher today's price level is, given the value of Mt-1. The important point is that under policy regime 2, the government prints money to meet its current Юnancing need. This translates a short-run need for Юnancing into inЯation, something that does not occur under regime 1. Under regime 1, the government, instead, meets its Юnancing needs through borrowing, and at some later date implements a Юscal reform that allows it to avoid using monetary Юnancing. Of course, in reality, there are all sorts of other policy regimes that Юt somewhere between our two polar cases. The important lesson from our simple analysis is that the time series correlation between deЮcits and inЯation depends critically on which policy regime we are in. In regime 1, deЮcits and inЯation are uncorrelated: inЯation is always zero no matter what the primary deЮcit is. In regime 2, primary deЮcits and the inЯation rate are strongly positively correlated. Policy Lessons A lack of correlation between deЮcits and inЯation might be naively interpreted as indicating that somehow inЯation is driven by something other than the government's Юscal policy, and that it has a life of its own. It is important that policy makers should not be misled into believing this. Even if it does not coordinate with the Юscal policy maker, the monetary authority can smooth the effects of Юscal policy on the price level by avoiding a monetary policy similar to (2.7). However it cannot prevent inЯation occurring if the government is Юscally irresponsible. To see this, notice that the central bank is always free to adopt a constant money growth rule, regardless of the government's choice for the path of xt. Notice that if ln Mt = µ, it is easy to show that Pt = eµ-aMt and, therefore, that ln Pt = µ. So the monetary authority can smooth inЯation, avoiding the Яuctuations in inЯation that would be inherent in policy regime 2. Despite its ability to smooth inЯation, however, the central bank cannot suppress it if the Юscal authority is irresponsible. With constant money growth, the Яow of real seigniorage 8
revenue is constant, allowing us to rewrite (1.9) as

=
rbt-1
-
r
! (1
+
r)-(i+1)xt+i
i=0
where

=
Mt
- Mt-1
=
ea-µ
%
&
1 - e-µ .
Pt
Notice that the less Юscally responsible the government is, the smaller $ i=0(1 + r)-(i+1)xt+i
is. This implies that the less Юscally responsible the government is, the larger must be,
and hence, the bigger µ and, thus, the more rapid, inЯation must be. A simple graph of the
relationship between and µ is found in Figure 1.
Thus, when we look across countries, we should expect to see higher inЯation in countries
with less Юscally responsible governments, as long as we measure inЯation and primary
balances over reasonably long horizons.
Fiscal and Monetary Policy Coordination In this section we extend the analysis of the previous section by considering the coordination of monetary and Юscal policy. In the previous section we saw that for a given country Юscal deЮcits and inЯation need not be correlated with one another if the monetary authority chooses a constant money growth rule. Conditional on such a policy, we saw that the Юscal authority's choice for the path of the primary surplus determines the money growth rate and the inЯation rate. It is clear that one choice for the government and the central bank is to coordinate policy. They could agree on a desired inЯation target, , and the central bank could set the money growth rate consistent with . The government, in turn, could ensure that its choice of the path for the primary surplus would be consistent with (1.9), given . Alternatively, it is interesting to considering the case of uncoordinated policy. In this case, like Sargent and Wallace (1981) we could imagine a Юscal authority that chooses {xt} t=0 without regard to any coordinated policy goals. The monetary authority, on the other hand, attempts to do what most central banks do: Юght inЯation. Initially the central bank Юghts inЯation by setting a low value of the money growth rate. Eventually, however, in the world Sargent and Wallace imagined, the central bank has to face the reality of the Юscal authority's dominance. It must eventually accommodate the government's Юnancing needs by printing more money. We are interested in the consequences of such a policy. To determine the implications of uncoordinated policy it is helpful to adopt the quantity
9
theory speciЮcation of money demand, so that Mt/Pt = yt/v.5 We will assume that the transactions motive for money demand is constant, i.e. yt = y, so that real balances are also constant: Mt/Pt = m = y/v. In this setting if money growth is constant at some rate µ, it is clear that the inЯation rate will be = µ, and that the real value of seigniorage will be constant: t = , where
= Mt - Mt-1 = (1 + µ)Mt-1 - Mt-1 =
µ m.
Pt
(1 + µ)Pt-1
1+µ
(2.8)
We will model the central bank's initial desire to be tough on inЯation by assuming that from period 0 to some period T , it sets the money growth rate to some arbitrary "low" value, µ. However, at date T , the central bank accepts the inevitable, that it will have to print more money to ensure the government's solvency. Therefore, from date T forward, the central bank sets the money growth rate to a constant, µ$, consistent with satisfying the government's lifetime budget constraint. With these assumptions we can easily solve for, and characterize, the path of inЯation for t 0. We will assume, for simplicity, that the Юscal authority sets xt = x for all t, and that x < rb-1. The second assumption implies that some seigniorage revenue will be required in order for the lifetime budget constraint to be satisЮed. Notice that the government's lifetime budget constraint as of period T + 1 is
bT = ! (1 + r)-(t-T )(xt + t). t=T +1
(2.9)
Since t = µ$m/(1 + µ$) for t > T , and xt = x for all t, this means that (2.9) can be rewritten
as
bT
=
x + µ$m/(1 + µ$) . r
(2.10)
Notice that if we solve for µ$ we obtain
µ$ = rbT - x . m - (rbT - x) Clearly the higher the level of debt at date T , the higher µ$ must be, since
(2.11)
dµ$
rm
dbT = [m - (rbT - x)]2 > 0.
5The results in this section, regarding the qualitative properties of the path of inЯation, hold true for more general money demand speciЮcations.
10
The budget constraint rolled forward from period 0 to period T is
b-1
=
(1
+
r)-(T +1)bT
+
! T (1
+
r)-(t+1)(xt
+
t)
t=0
which we can rewrite as
"
#
bT
=
(1 + r)T +1b-1
-
(1 + r)T +1 r
-1
x
+
m
1
µ +
µ
Notice that
'
"
#(
bT T
= ln(1 + r)(1 + r)T+1
1 b-1 - r
µ x + m1 + µ
,
which is positive if the central bank sets µ low enough. Also
(2.12)
bT
=
-
(1
+
r)T +1
-
1 m(1
+
µ)-2
<0
µ
r
(2.13)
So the government accumulates more debt with lower µ. Together these results imply that the tougher the monetary authority is initially (the lower it sets µ) and the longer it is tough (the higher is T ), the greater the stock of debt (bT ) will be when it Юnally recognizes the necessity of satisfying the government's need for Юnancing. But the greater the stock of debt bT , the higher the inЯation rate will eventually be. The basic policy message that emerges from this discussion is that, the tougher the monetary authority tries to be on inЯation in the near term, the higher inЯation will be in the long term. Having stably low inЯation requires the coordination of Юscal and monetary policies.
Getting InЯation Under Control The discussion so far has described inЯation as a
problem that stems from loose Юscal policy. In particular, when the government sets its
primary balance too low, so that
! (1
+
r)-(t+1)xt
%
b-1
t=0
the central bank is forced, at some point, to print money. The logical consequence of printing
money is inЯation. This suggests that in order for the government to reduce inЯation it must
impose some degree of Юscal discipline. In a world with uncertainty, the government must
11
not only impose Юscal discipline, it must convince other agents that it will remain disciplined in the future.6 Policy makers frequently argue that inЯation stems from other root causes, and that it is very difficult for them to eliminate inЯation once it becomes part of people's everyday lives. One often hears of the role played by private expectations. While it is possible to construct examples in which outcomes can depend on self-fulЮlling changes in agents' expectations, the role played by expectations is often over-emphasized. As Sargent (1983) argues, the fundamentals, namely Юscal policy, are often the important determinant of private expectations. In particular, if the government announces a credible policy regime shift that involves a shift to a permanently better primary balance, inЯation can be brought down, and it can be brought down quickly. Sargent's argument is based on the shared experiences of Austria, Hungary, Poland and Germany after World War I. All four countries ran large deЮcits after the war, and experienced hyperinЯations. All four countries used Юscal rather than strong monetary measures to end their hyperinЯations. They renegotiated war debts and reparations payments, which represented a substantial part of their Юscal burdens, and they announced other Юscal measures to contain their budget deЮcits. However, even after the hyperinЯations were over, the four governments continued, for several months, to print money at a healthy pace. By following this policy mix all four governments quickly stabilized their price levels and exchange rates. The lesson from these case studies is straightforward. In the face of credible Юscal reforms, private expectations of future money growth will adjust and the price level and inЯation will stabilize. While the example of the central European economies in the interwar period may not be perfectly analogous to today's developing and emerging market economies, the story is still valuable. If private expectations of inЯation are entrenched in these markets, why are they entrenched? Our theory suggests that it must stem from Юscal difficulties. Our theory also suggests that without fundamental steps being taken to correct Юscal imbalances, no monetary policy that attempts to control inЯation can be successful indeЮnitely. 6As we have seen with the Cagan money demand examples, the current price level depends on current and expected future money supplies. Hence, even if the government eliminates its used of monetary Юnancing today, it must convince other agents that it will avoid monetary Юnancing in the future. 12
3. The Fiscal Theory of the Price Level A branch of macroeconomic theory that has recently become more popular, the Юscal theory of the price level, differs in its interpretation of the government's lifetime budget constraint. In fact, this theory would not even admit that the equation (1.9) represents a constraint on the government.
Interpreting the Lifetime Budget Constraint Let us return to (1.9) again, and consider how we obtained it. We started from the Яow budget constraint, (1.3), which represents a simple accounting identity that holds under certain assumptions about the real interest rate and the structure of debt. From (1.3) we derived an intertemporal equation, (1.7). If we imposed the condition, (1.8), which we repeat here as
lim (1 j
+
r)-(j+1)bt+j
=
0,
(3.1)
then the lifetime budget constraint, (1.9), emerged.
Equation (3.1) is often referred to as a no-Ponzi scheme condition on the government. So
imposing the lifetime budget constraint on the government's behavior is often thought of as
being equivalent to not allowing the government to run a Ponzi scheme. However, as McCal-
lum (1984) points out, if we consider theoretical settings in which optimizing households are
the potential holders of government debt, any violation of the government lifetime budget
constraint, (1.9), implies either that the households are not optimizing, or that they are
violating a no-Ponzi scheme condition imposed on them. As Cochrane (2001b) points out,
the transversality, or no-Ponzi scheme, condition that we are familiar with from dynamic
consumer problems is applied to households so that dynamic trading opportunities do not
broaden the household budget set relative to having a single contingent claims market at
time 0. So, it would appear that an additional constraint on the government is not needed
to justify (1.9). And it would appear that this condition is the consequence of imposing
rationality, and a no-Ponzi scheme condition on households.
To see this worked through suppose that we let xt + t = -k, so that debt increases in
each period by an amount
bt - bt-1 = rbt-1 + k.
(3.2)
If k > -rb0, then debt grows over time with
lim (1 t
+ r)-tbt
=
b0
+
k r
>
0.
(3.3)
13
In simple dynamic macroeconomic models this would immediately imply a violation of the type of transversality (or no-Ponzi scheme) condition that we usually impose on households. Importantly, however, allowing the government's debt to grow fast enough that (1.8) is violated also implies a violation of optimizing behavior on the part of households. Notice that the household's Яow budget constraint will usually look like
income - purchases = (1 + r)-1bt - bt-1
(3.4)
Notice that, in our example, this means
income - purchases = k
Clearly this means the household is doing something suboptimal, since it is voluntarily giving up a constant stream of income that could be used to permanently increase its consumption. In essence, this is the nature of argument in McCallum that we need not think of the government budget constraint as an additional constraint on the government. Cochrane (2001b) presents a much more general argument than McCallum's. His argument is more complex, but makes clear why we can think about the government budget constraint as an equation for valuing government debt, rather than as a budget constraint. In the next section we will consider a similar, but simple argument, put forward by Christiano and Fitzgerald (2000).
Fiscal Theory in a Nutshell Christiano and Fitzgerald (2000) use a simple one-period model to describe the Юscal theory of the price level. Their model assumes that the world lasts for just one period. Households enter the period with real claims on the government, b. Households demand end-of-period claims on the government, b$. Clearly households will not pick b$ > 0 because the world ends at the end of period 1. The argument is simple. Any household, no matter what the exact speciЮcation of its budget constraint is, that demands b$ > 0 is wasting resources­in essence this is the same point being made by McCallum (1984) in the context of a dynamic model. If households were completely unconstrained in their choice of b$, they would pick b$ = -. So usually we impose b$ 0. That is, we do not allow households to end life with unpaid debt. This constraint, in a one-period model, is the natural analog to the no-Ponzi scheme condition in a dynamic inЮnite horizon model. Since optimizing households will not
14
choose b$ > 0 and we constrain them to have b$ 0, it is clear that households will pick b$ = 0 no matter what government policy is in the period. In a one-period model, the government's budget constraint is simply
b$ = b - x -
(3.5)
because b now represents total claims on the government, i.e. it represents principal plus interest. This is just a notational difference, and is of no consequence for the arguments being made. The fact that households will pick b$ = 0 for any combination of government policies implies that the government budget constraint is
b = x + .
(3.6)
Real Debt Suppose we are in the world similar to the one we described earlier, where government debt is a claim to a real quantities of goods. This means the value of b is Юxed in real terms. This means that if the government chooses a "loose" Юscal policy (i.e. in the sense that x % b) it is clear than the monetary authority must provide the necessary "loose" monetary policy ( ' 0) to Юnance the government. Here we have the arguments made in the previous section of this chapter in an incredibly simple form. If the government sets Юscal policy without regard to the price level or rate of inЯation, the central bank must accommodate by printing money. When we saw this before in a dynamic setting, there was an issue of timing. The central bank faced a choice between inЯation now or later, but the central bank could not avoid printing money eventually in the face of loose Юscal policy.
Nominal Debt Now imagine, instead, that, as is the case for most of the debt issued by the United States government, the government's debt represents a claim to a certain number of units of local currency. Now the government budget constraint is
B$ = B - P (x + ).
(3.7)
Since, again, households will not want to hold government debt at the end of the period, we will have B$ = 0. This implies B = P (x + ), or
B/P = x + .
(3.8)
15
Notice, now, that the government and the central bank are free to choose any x + combination subject to x + > 0. Given a policy commitment to x and , a beginning of period market for government debt will induce a price level P that satisЮes the "government budget constraint." In this sense, the government is not constrained in its actions and (3.8) does not represent a government budget constraint. Instead, (3.8) provides a way of valuing the nominal government debt issued in the previous period.
Is the Fiscal Theory Relevant? The curious reader might wonder whether the Юscal theory of the price level has any relevance to real world policy making. The astute reader might also wonder how we can have two equations determining the price level: the lifetime budget constraint and the solution for the price level derived from a money demand speciЮcation. The answers to these two questions are related. The basic relevance of the Юscal theory is established by the fact that many governments issue nominal debt. But the fact that the Юscal theory does not overdetermine the price level depends crucially on the assumption that some monetary policy rules do not specify an exogenous path for the money supply. In particular, according to some monetary policy rules, the money supply is endogenous. Consider, for example, the case where money demand conforms to the Cagan speciЮcation we saw above, so that Mt = Ayt exp(-Rt)Pt, where yt is real output, Rt is the nominal interest rate and Pt is the price level. Suppose that output is constant, yt = y and that the central bank pegs the nominal interest rate at some value R. Clearly this policy requires that whatever the price level, Pt, the central bank must ensure that the monetary base is equal to Mt = Aye-RPt. But notice that since this rule speciЮes the money supply in terms of the price level, this policy rule does not allow us to solve for the price level using (2.3). This is where the government's lifetime budget constraint comes in. Notice that real balances are constant and equal to m = Aye-R. If the real interest rate is some constant, r, the central bank's peg of the nominal interest rate also implies that the inЯation rate must be constant, and equal to = (R - r)/(1 + r). This implies that the real value of seigniorage is

=
Mt
- Mt-1 Pt
=
Mt Pt
-
1
1 Mt-1 + Pt-1

=
m.
1+
(3.9)
16
If we return to the dynamic model where (1.9) must hold notice that we now have
bt-1
=
! (1 i=0
+
r)-(i+1)xt+i
+
1 r
1
+
m.
If the government's debt at the beginning of time t is denominated in units of local currency,
and equals Bt-1, this means that
Bt-1 Pt
=
! (1 i=0
+
r)-(i+1)xt+i
+
1 r1
+ m,
(3.10)
as long as there is no uncertainty beyond date t. Notice that since and m are determined by the central bank's peg of the nominal interest rate, this means the price level is determined by the present value of the stream of future primary surpluses plus seigniorage, relative to the quantity of government debt in circulation. The money supply can then be obtained by multiplying Pt by m.
Does the Fiscal Theory Lead to a Different Policy Message? The short answer to this question is no. The Юscal theory still sends the message that government deЮcits are inЯationary if the government does not explicitly default on debt obligations. The only distinction between the Юscal theory and the earlier literature that we discussed above is that inЯation can happen in one of two ways. First, consistent with the discussion where debt was real, inЯation could result from the central bank using a money supply rule that preserves the real value of the government's debt at the time it was issued. Second, inЯation could result from the government issuing nominal debt the real value of which it does not attempt to preserve. In the face of a Юscal shock that reduces the present value of the government's future primary surpluses, the price level might jump in order that (1.9) holds.
4. Conclusion In this opening chapter we have derived the government's lifetime budget constraint under some simple assumptions about government debt: time is discrete, debt is issued for one period, is real, and bears a constant real interest rate. We showed that under these assumptions Юscal sustainability revolves around the Юscal and monetary authorities setting the paths of the primary surplus, xt, and the supply of base money, Mt, consistent with
17
(1.9). We argued that there are many combinations of Юscal and monetary policy consis-
tent with (1.9). However, we argued that the inevitable consequence of loose Юscal policy,
$ t=0
(1
+
r)-(t+1)xt
% b-1, is inЯation.
The monetary authority cannot Юght inЯation in-
deЮnitely without the cooperation of the Юscal authority. Thus, the goal of low inЯation,
combined with Юscal sustainability, can only be achieved if monetary and Юscal policy are
coordinated. This message does not change when we consider alternative interpretations of
the government's lifetime budget constraint, such as those provided by the Юscal theory of
the price level.
18
References Christiano, Lawrence J. and Terry J. Fitzgerald (2000) "Understanding the Fiscal Theory of the Price Level," manuscript, Northwestern University. Cochrane, John H. (2001a) "Long-term Debt and Optimal Policy in the Fiscal Theory of the Price Level," Econometrica, 69, 69--116. Cochrane, John H. (2001b) "Money as Stock," manuscript, University of Chicago. Corsetti, Giancarlo and Bartosz Mackowiak (2002) "Nominal Debt and the Dynamics of Currency Crises," forthcoming, Journal of International Economics. Daniel, Betty (2001) "A Fiscal Theory of Currency Crises," International Economic Review, 42, 969--88. Dupor, William (2000) "Exchange Rates and the Fiscal Theory of the Price Level," Journal of Monetary Economics, 45, 613--30. Hamilton, James D. and Marjorie A. Flavin (1986) "On the Limitations of Government Borrowing: A Framework for Empirical Testing," American Economic Review, 76(4), 808--19. McCallum, Bennett T. (1984) "Are Bond-Юnanced DeЮcits InЯationary? A Ricardian Analysis" Journal of Political Economy, 92, 123--35. Sargent, Thomas (1983) "The Ends of Four Big InЯations," in Robert E. Hall, ed., InЯation: Causes and Effects, 41--97. Chicago: University of Chicago Press for the NBER. Sargent, Thomas (1985) "Reaganomics and Credibility," in Albert Ando, et. al., eds. Monetary Policy in Our Times, 235--52. Cambridge, MA: MIT Press. Sargent, Thomas and Neil Wallace (1981) "Some Unpleasant Monetarist Arithmetic," Federal Reserve Bank of Minneapolis Quarterly Review, 5(3), 1--17. Sims, Christopher (1994) "A Simple Model for the Determination of the Price Level and the Interaction of Monetary and fiscal policy," Economic Theory, 4, 381--99. Woodford, Michael (1995) "Price Level Determinacy Without Control of a Monetary Aggregate," Carnegie-Rochester Conference Series on Public Policy, 43, 1--46. 19
FIGURE 1 SEIGNIORAGE AS A FUNCTION OF THE MONEY GROWTH RATE rb Annuity value of future primary surpluses Seigniorage flow µ Note: Under the assumptions listed in the text, the constant seigniorage flow is , given a constant money growth rate µ.

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