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We would like to thank Tanya Kirsanova, Campbell Leith, Warwick McKibbin, John Muellbauer and Simon Wren-Lewis, and participants at seminars at Oxford University, the Australian National University, the Bank of England and the International Monetary Fund.
Fiscal rules in the UK consist of the ‘Golden Rule’, which restricts fiscal policy from borrowing over the cycle other than to finance investment, and the ‘Sustainable Investment Rule’ which requires public debt to remain below the ‘prudent’ level of 40% of GDP.
Notice that the idea that the government can use inflation surprises to reduce the real value of debt, and hence behave in a time-inconsistent manner, goes back to Lucas and Stokey (1982), Persson et al (1987) and Calvo and Obstfeld (1990).
Clarida et al (1998) is one study amongst many, which finds that the Taylor principle has been fulfilled by, for example, the US, Germany, Japan and the UK over the last two decades.
The requirement that the household’s wealth accumulation satisfies the transversality condition is given by lims→∞ Et (Rt,sĀs) = 0.
The choice of B in turn determines the steady-state tax rate τ.
The forward-looking elements of the system will leave this finding unchanged as monetary policy will become more effective in both consumption and output control.
To see this, we take the limit of (15) for ω → 1. We substitute out using the Phillips curve in this limit, which Steinsson (2003) has shown equals πs = πs-1 + δ (1 − γ)yt−1, and obtain limω → 1 (1 − ω) Ws = 0.
The weights Ω23 and Ω32 are non-zero because they constitute the weight on the squared output gap which we substituted out for in terms of consumption and government spending using (13).
We also notice from (28) that
Woodford (2003b) has suggested an alternative solution approach in which the system, in all periods including the initial, follows the first order conditions for s > 0. This ‘timeless perspective’ policy is not time inconsistent as it involves ignoring the conditions that prevail at the regime’s inception (s = 0). We want to analyse the time inconsistency inherent in commitment policy and will therefore consider fully optimal commitment policy.
Levine (1988) has shown that the diagonal entries of Φ22 are negative and hence that the incentive to renege exists at all points along the trajectory path of optimal commitment policy.
This calibration is chosen to correspond approximately to χf = χb = 0.5, as in Fuhrer and Moore (1995).
The maximum eigenvalue of a system describes the speed of adjustment of the system and hence that of its most persistent process.
These optimal coefficients under commitment have to be interpreted with care as the fully optimal rule includes feedback onto the pre-determined Lagrange multipliers (see (31)).
From (31) we see that under commitment the first period value of the instrument is given by (i1, g1)′ =
where we have substituted for the unit shock µ1 = 1, the initial conditions π0 = y0 = b1 =0 because we start from equilibrium, and ρπ = ρc = 0. Under discretion it follows from (36) that we have
Given the setup of the model, the condition
Reducing debt is necessarily inflationary in this setup because it will be done, to a large extent, by lowering interest rates. That is both because inflation helps to reduce debt directly and because reducing debt only by lowering government spending would be costly, because the level of government expenditure features in the utility function.
We notice that as in Kirsanova and Wren-Lewis (2007) the optimal fiscal feedback on debt is negative and small.
The initial linearisation remains valid despite this shift in steady state if this change is small in magnitude.
We further notice that inflation falls below zero and then rises back to zero. For the same reasons as just discussed, it becomes optimal for iscal policy to raise spending to align the rule-of-thumb price setters with the Calvo price setters when inflation is negative.
It follows that this incentive to reduce debt through inlation is higher if debt is denominated in nominal terms, rather than in real terms as in the present model. See Leith and Wren-Lewis (2007).
This discussion implies that policy under fully optimal commitment policy will induce slightly higher inflation and lower debt in the first period as compared with ‘timeless’ commitment policy, see Leith and Wren-Lewis (2007).
A smaller incentive to renege under commitment does not mean, however, that the requirement to conduct time-consistent control of debt will imposes a smaller distortion onto the system. We will return to this issue below.
That is, we observe a smaller percentage deviation of debt from its steady-state level.
We see in Column (3) of Table 1 that the fiscal debt feedback rises in absolute value from −0.005 to −0.027.
We also see this by observing that B does not feature in (29), which is the source of the time inconsistency problem in inflation control.
An argument by reductio ad absurdum makes this point clear. Any candidate for a discretionary outcome which had a positive outcome for debt would be vulnerable, at any point after this supposed equilibrium had been reached to re-optimisation by the policymaker to reduce debt, taking inflation expectations as given. But this vulnerability would cause the candidate equilibrium to unravel, by backwards induction.
Notice how interest rates help to accumulate less debt. Whilst interest rates rise more strongly under discretion they are returned much more quickly to zero than under commitment. This lower cumulative effect of interest rates helps to limit the accumulation of debt.
That is, the policymaker under discretion can still make promises about future policy but these promises have to be time consistent.
Notice, however, that this cut in spending is not necessary to control inflation in the first period. Stehn (2007) shows that interest rates under optimal discretionary policy continue to be cut initially even if government spending is unable to fall because fiscal policy is constrained to a simple feedback on debt (and hence cannot respond to the cost-push shock in the first period).
Notice that we would expect this violation of the Taylor principle to disappear for an entirely backward-looking inflation process, as cutting the interest rate would lead to an explosive inflation process. However, as the micro-founded social loss function (15) is not defined in this limit, we cannot compute optimal policy. Kirsanova et al (2005), for a non-microfounded model, show that monetary policy fulfils the Taylor principle with an accelerationist Phillips curve, regardless of the level of steady-state debt.
For example, whilst Euroland has a debt to GDP ratio about 60%, the amount of debt re-financed per year is much lower at around 11% of GDP (ECB Monthly Bulletin 2005).
For example, Steinsson (2003) shows that the absolute value of the welfare loss under discretionary policy is 26% higher than under commitment for a New Keynesian Phillips curve in a monetary policy model. In our model with public debt, this loss rises to over 50%.