Economics Division University of Southampton Southampton SO17 1BJ, UK

Discussion Papers in Economics and Econometrics

Title: Nominal GDP Targeting and the Tax Burden

Authors : Michael Hatcher (University of Southampton) No. 1510

This paper is available on our website http://www.southampton.ac.uk/socsci/economics/research/papers

ISSN 0966-4246

1

Nominal GDP targeting and the tax burden

Michael Hatcher

Department of Economics

University of Southampton, SO17 1BJ

Email: M.C.Hatcher@soton.ac.uk.

14 Dec, 2015

Abstract

In an economy with nominal government debt, nominal GDP targeting has two opposing

effects on the excess tax burden. On the one hand, taxes are higher on average than under

inflation targeting; on the other, taxes are less volatile. Numerical analysis shows that the

average excess burden is lower under nominal GDP targeting because the tax volatility effect

dominates. Under inflation targeting, the excess burden of taxes is minimised by issuing only

indexed government debt. The same is not true under nominal GDP targeting.

Keywords: nominal GDP targeting, inflation targeting, government debt, tax burden.

JEL classification: E52.

1 Introduction

Modern monetary policy analysis has focused mainly on the role of monetary policy as a tool

for minimizing inefficient price variations across firms due to nominal price rigidities.1

However, this is not the only mechanism through which monetary policy matters. Recent

research has emphasised that unanticipated inflation has real wealth effects on households

who enter into nominal financial contracts but lack full insurance against inflation risk (e.g.

Doepke and Schneider 2005, 2006). In these circumstances, the choice of monetary policy

regime is non-trivial. Despite this, little is known about the relative merits of inflation

targeting and other regimes in this context. 2 This paper takes a step in this direction by

assessing the impact of nominal GDP targeting on the tax burden.3 To do so, an overlapping

generations model is presented in which the key ingredient is nominal government debt.

The analysis is motivated by the observation that nominal GDP targeting implies a

countercyclical price level, so that bondholders are hit with surprise inflation at times when

aggregate income is low. Since risk-averse bondholders must be compensated for this risk,

government borrowing costs increase relative to inflation targeting, raising average taxes. At

the same time, however, taxes are less volatile under a nominal GDP targeting regime. The

overall impact on the excess tax burden depends on whether the tax level effect dominates the

tax volatility effect. Numerical analysis indicates that the average excess burden is lower

under nominal GDP targeting because the tax volatility effect dominates. An additional

finding is that, under inflation targeting, the average excess burden of distortionary taxes is

minimised by issuing only indexed debt. By contrast, nominal GDP targeting calls for

issuance of mainly nominal government debt in order to minimise the excess tax burden.

1 See, for example, Woodford (2003). 2 Exceptions include Meh, Rios-Rull and Terajima (2010), Koenig (2013) and Sheedy (2014). 3 Since no central bank has adopted nominal GDP targeting in practice, previous academic studies of nominal

GDP targeting have been theory-based rather than empirical.

2

The paper is related to two main strands of literature. First, there is a past literature on

nominal GDP targeting. Formal analyses include Bean (1983), Bradley and Jansen (1989)

and Hall and Mankiw (1993), amongst others. These papers highlight circumstances in which

nominal GDP targeting can be expected to increase social welfare. More recently, Billi

(2013) extends the analysis to the New Keynesian model. He finds that nominal GDP

targeting can be beneficial in the presence of the zero lower bound on nominal interest rates.

A somewhat different issue is studied by Koenig (2013) and Sheedy (2014). They show that

if private financial contracts are specified in nominal terms, stabilising nominal GDP

redistributes income from borrowers to lenders and can replicate the efficient allocation in the

presence of complete financial markets. The analysis here contributes to the nominal

contracting part of the literature by studying the implications of nominal GDP targeting for

government borrowing costs and the excess burden of distortionary taxes. This contrasts with

previous work in the literature, which has studied the effects of nominal GDP targeting in the

presence of private nominal debt and wage contracts, but not government debt.4

The paper is also related to a separate literature on the redistributive effects of inflation

through changes in the real value of nominal government debt. In a seminal paper, Doepke

and Schneider (2006) show that US households have substantial net nominal debt positions

and that episodes of unanticipated inflation lead to substantial wealth redistribution across

households.5 Government debt plays a crucial role in this redistribution because the old are

the main bondholders in the economy, as in the model presented here. Theoretical models of

the redistributive effects of inflation via the government debt channel include Champ and

Freeman (1990), Doepke and Schnieder (2005) and Meh, Rios-Rull and Terajima (2010).

Like the present paper, these studies use an overlapping generations model. The novel feature

here is that monetary policy affects average government borrowing costs and the average

level of distortionary taxes because the effects of inflation risk are taken into account. This is

crucial since nominal GDP targeting raises inflation risk and thus affects average government

borrowing costs. A sound evaluation of the impact of nominal GDP targeting on the tax

burden requires that this effect be taken into account. This paper provides such an analysis.

The remainder of the paper proceeds as follows. Section 2 presents the model, calibrates it

and discusses the effects of nominal GDP targeting on the tax burden. The main results are

reported in Section 3, followed by a discussion of extensions and robustness in Section 4.

Finally, Section 5 concludes.

2 Model

The model consists of a small open economy with an infinitely-lived government and

overlapping generations of representative households that live for two periods. Consumption

when young (old) is denoted c1 (c2). The young receive an endowment income equal to a

fraction 1 – ε of aggregate output, Y. They are taxed by the government in the amount τ,

which is taken as given. Taxes are assumed to be distortionary, with excess burden f(τ) which

4 In practice, most government debt is nominal. For instance, indexed (i.e. real) bonds were around 7% of

marketable Canadian government debt in 2011 (Department of Finance Canada, 2011) and 10% (30%) of US

(UK) marketable government debt in 2008 (Campbell, Viceira and Shiller, 2009). 5 Similar results are obtained for Canada (Meh and Terajima, 2011) and the Euro Area (Adam and Zhu, 2015).

3

satisfies f(0) = 0, fτ(τ) > 0 and fττ(τ) > 0.6 The other fraction of aggregate output, ε, is received

by the old. The natural log of output follows an AR(1) process with conditional variance σY2.

Young agents can save for old age by investing in nominal government bonds, bn, or indexed

government bonds, bi. Nominal bonds pay a riskless nominal interest rate R on maturity. The

ex post real return on nominal bonds is given by Rt/Πt+1, where Πt+1 ≡ Pt+1/Pt is inflation, and

Pt is the aggregate price level. Indexed bonds pay a riskless real return rf > 1 which

coincides, due to arbitrage, with the constant world real interest rate. 7

A young agent born at date t solves the following problem:

)()( max 1,2,1,

tttbb

cUEcUit

nt

s.t. i

t

n

tttttt bbYfYc )]([)1(,1 (1)

i

tft

n

ttttt brbRYc ,111,2 )/(

where β > 0 and households of ages j = {1,2} have CRRA utility

1

,,1

1)( tjtj ccU (2)

with coefficient of relative risk aversion γ > 0.

The first-order conditions for bonds are as follows:

])([)( 1

11,2,1

ttctttc cUERcU (3)

)]([ )( 1,2,,1 tctfttc cUErcU (4)

The government issues debt and sets taxes to cover fixed real spending g* > 0 plus interest

payments on debt net of new issuance. The government budget constraint is given by:

tttttftt

i

t

i

tft

n

t

n

ttttt

bbRvrvg

bbrbbRgY

11,1

1,111

)]/)(1([*

)/(* (5)

where b = bn + bi is total supply of government debt and v ≡ bi/b is the share of indexed debt.

In what follows, we consider equilibria in which the share of nominal debt is held constant.

Therefore, the supply of nominal debt is given by btn = (1 – v)bt and the supply of indexed

debt is bti = vbt. Market-clearing requires that demand for each type of debt equal its supply.

2.1 Monetary policy

As in Koenig (2013) and Sheedy (2014), monetary policy determines inflation directly.

However, the central bank has only imperfect control of the price level. In particular, the

6 Bohn (1988) models the excess burden this way. The main advantage of this specification is that it allows the

tax burden to be decomposed into mean and volatility components: a rise in mean taxes raises the excess burden,

and so does a mean-preserving increase in tax volatility. Lump sum taxes are recovered by setting f(τ) = 0. 7 Domestic households are indifferent between home and foreign bonds. It is assumed that all household demand

for bonds is satisfied by the home government, so that demand for foreign bonds is zero. This is a simplifying

assumption and does not change the main conclusions; see Section 4.3 for further details.

4

price level equals its desired value P* multiplied by a control error exp(εt), where εt is IID-

normal with mean zero and standard deviation σε. Hence, Pt = Pt

*exp(εt) under both regimes.8

Under inflation targeting there is base-level drift, so the target price level is given by

Pt*= Π*Pt-1. Hence, inflation is equal to

)exp(* t

IT

t (6)

Under nominal GDP targeting, the desired price level Pt* satisfies Pt

* Yt = PY*t, where

PY*t = [Π*(1+g*)]t is the nominal GDP target in period t, and target output growth is

g* = 0.9 Inflation under nominal GDP targeting is therefore given by10

)exp(/)exp(]/[* 11 tttt

NIT

t YY (7)

Nominal GDP targeting implies a countercyclical price level: it makes inflation and output

negatively correlated. By contrast, there is no relationship between the price level and output

under inflation targeting, so that the only source of inflation risk is the control error εt.

The inflation variances under the two regimes are related as follows:

2

11 )(lnvar)(lnvar Y

IT

tt

NIT

tt (8)

where 2

1 )(lnvar

IT

tt .

Intuitively, inflation is more volatile under nominal GDP targeting because it responds to

fluctuations in real GDP. As noted later on, this increase in inflation risk has implications for

the inflation risk premium, and hence government borrowing costs and average taxes.

2.2 Equilibrium

Definition of equilibrium:11

A set of allocations and prices fttt

sn

t

dn

t

si

t

di

tttt rRbbbbcc ,

,,,,

,2,10 ,,,,,,,,

with the following

properties for all t:

(1) Allocations sn

t

dn

t

si

t

di

ttt bbbbcc ,,,,

1,2,1 ,,,,, solve the utility maximisation problem of the

young born at date t;

(2) The goods and bonds markets clear:

)](1[*,2,1 tttt fYgcc

si

t

di

t bb ,,

sn

t

dn

t bb ,,

8 The absence of money simplifies the analysis but does not affect the main conclusions (see Section 4.2). 9 It is assumed, without loss of generality, that the initial nominal GDP target, PY*0, is equal to one. 10 The response of inflation to the lagged policy error εt-1 reflects the fact that past deviations from the nominal

GDP target are offset under nominal GDP level targeting. Under nominal GDP growth targeting there is no

response to the lagged policy error but essentially identical results are obtained. 11 Note that d and s superscripts are introduced in this section to denote demand and supply values. These

superscripts are omitted in other sections of the paper in order to avoid unnecessary notation.

5

(3) The government budget constraint holds: si

t

si

tft

sn

t

sn

ttttt bbrbbRgY ,,

1,1

,,

11 )/(*

(4) The real interest rate on indexed bonds equals the constant world interest rate: *, rr ft

2.3 Monetary policy and the tax burden

Nominal GDP targeting has two opposing effects on the excess burden of taxes, f(τ). This is

because taxes are higher on average under nominal GDP targeting, but also less volatile. In

this subsection, these two effects are described in greater detail before turning to the problem

of minimising the average excess tax burden.

Level effect: average taxes

Nominal GDP targeting raises mean taxes because it implies an increase in the inflation risk

premium on government debt. This, in turn, pushes up average borrowing costs.12 We can

illustrate this mechanism quite simply using the model at hand. Using (2) and a second-order

approximation of (3) and (4), the inflation risk premium is given by

]ˆ,ˆ[cov

ˆ]ˆ[var5.0ˆ

11,2

,1,1

ttt

ftttntt

c

rrEIRP

(9)

where rn denotes the real return on nominal debt, ‘hats’ denote log deviations from the

deterministic steady state and the inflation variance term arises due to Jensen’s inequality.

Using (5) and (6)-(8), the inflation risk premia under inflation and nominal GDP targeting are

2

2

)1(

c

bvrIRP

nIT (10a)

IT

Y

nNIT IRP

c

bvrYIRP

][

)1( 22

2

(10b)

The inflation risk premium is higher under nominal GDP targeting for two reasons. First,

since an unexpectedly low level of output implies surprise inflation, the correlation between

inflation and marginal utility is stronger than under inflation targeting. Second, inflation risk

is higher under nominal GDP targeting because the price level responds to fluctuations in

income; see Equation (8). If risk aversion is not too small, the increase in the inflation risk

premium will raise expected borrowing costs: E[rn,NIT] > E[rn,IT].13 By (5), this suggests that

nominal GDP targeting will raise average distortionary taxes for a given path of debt

issuance. This intuition is confirmed in the numerical analysis that follows.

Volatility effect: variance of taxes

The second effect of nominal GDP targeting on the tax burden comes from the fact that it

makes taxes less volatile. To see this, consider the government budget constraint (5).

Dividing through by output and using (6), the tax rate under inflation targeting is

12 For a survey of the literature on the inflation risk premium, see Bekaert and Wang (2011). 13 From (9) and 0ˆ

, ftr , IT

ntt

NIT

ntt rErE ,,,1,1ˆˆ if ])ˆ[var]ˆ[(var5.0 11

IT

tt

NIT

tt

ITNIT IRPIRP . By (10a) and

(10b), a sufficient condition is γ > 0.5c2(εY+rn (1– v)b)-1 , which holds under standard calibrations.

6

t

t

t

tt

t

t

ft

t

IT

tY

bb

Y

Rv

Y

rv

Y

g

1

1,1

)exp(*)1(

*

(11)

Following similar steps, the tax rate under nominal GDP targeting is

t

t

t

tt

tt

t

ft

t

NIT

tY

bb

Y

Rv

Y

rv

Y

g

1

1

11,1

)exp(*

)exp()1(

*

(12)

The first reason that taxes are less volatile under nominal GDP targeting can be seen from a

direct comparison of (11) and (12). Note that current output does not appear in the

denominator of the real return on nominal debt in (12). This effect makes taxes less volatile

under nominal GDP targeting for a given path of debt issuance because the real debt burden

is stabilised. Intuitively, since output equals the tax base, the tax rate has to vary inversely

with output shocks in order to maintain real government spending. However, since nominal

GDP targeting makes the debt burden positively related to current output, it induces a

positive correlation between the tax base and total outlays, so that taxes do not need to adjust

as much in the face of output shocks.

The second reason taxes are less volatile under nominal GDP targeting is that the demand for

government debt is more stable. This can be understood from the first-order condition for

indexed debt (4), or rf = β-1Uc(c1,t)/EtUc(c2,t+1). For this equation to be satisfied, the expected

stochastic discount factor must be stabilised perfectly, since the return on the left hand side is

constant. This requires large movements in the demand for debt unless monetary policy does

some stabilisation. This does not happen under inflation targeting because inflation moves

randomly. However, under nominal GDP targeting, the inverse relationship between inflation

and output implies some stabilisation of the expected stochastic discount factor, since the old

are exposed to more income risk (through bond returns) and the young to less (because their

income is taxed). This risk sharing implies that (4) holds for a more stable path of

government debt. By the government budget constraint, (5), this reduces tax volatility.

Minimising the excess tax burden

Since c1,t + c2,t = Yt[1 – f(τt)] – g*, the excess burden of taxes reduces the amount of goods

available for private consumption. It is therefore important to consider how the excess burden

can be minimised. This section and the numerical analysis that follows investigates this issue.

Formally, the burden-minimising debt share solves the following problem:

)]([ min]1,0[

tv

f E

(13)

s.t. (1)-(5), equilibrium conditions (see Section 2.2), and

NITunder )exp(/)exp(]/[*

ITunder )exp(*

11 tttt

t

tYY

where E is the unconditional expectations operator.

The debt share that satisfies (13) is computed numerically. To do so, the model was solved

using a second-order approximation in Dynare (see Adjemian et al., 2011). In particular, the

7

unconditional expectation of the tax burden was computed for a discrete number of debt

shares in the interval [0,1] by looping over the parameter v in small steps.

To understand the numerical results that follow, it is instructive to consider a second-order

approximation of the expected tax burden around the point τt = E[τt]:

effect VolatilityeffectMean

]var[ ])[()]([

fEffE (14)

where f~

is the second derivative of f(τ) evaluated at τ = E[τ].

This expression shows that the expected tax burden increases with mean taxes and tax

volatility. These moments are therefore reported in the numerical analysis. In addition, a

decomposition of the expected tax burden into mean and volatility effects is provided.

2.4 Calibration and steady state

In this section, the baseline calibration of the model is reported along with the deterministic

and stochastic steady states. The model is roughly calibrated to the UK economy

2.4.1 Calibration

Aggregate risk

The model contains two sources of risk: output shocks and monetary policy control errors.

Aggregate output follows an AR(1) process of the form:

ttYt eYY 1lnln (15)

where 10 Y and et is an IID-normal innovation with mean zero and variance σ2.

The parameter ρY was set at 0.7 as there is no convincing evidence that output is highly

persistent over generational horizons. The innovation standard deviation was set at σ = 0.035

since this implies a conditional variance of log output close to the 3.9% standard deviation of

real output growth from 1990-2015 based on overlapping 20-year sections from the

Production Index measure of output published by the ONS (GVA, chained volume).

The standard deviation of the policy error was set at σε = 0.05 because this implies an

inflation standard deviation of 7.5% per period under an inflation targeting regime. By

comparison, the 20-year standard deviation of UK inflation over the period 1988 to 2014 is

around 7% based on overlapping 20-year sections from the Consumer Prices Index (CPI).14

Distortionary taxes

The excess burden of distortionary taxes is assumed to be quadratic: f(τ) = (ϕ/2)τ2, where

0 < ϕ < 1. Here, ϕ controls the extent to which taxes are distortionary, with lump-sum taxes

recovered if ϕ = 0. The baseline calibration sets ϕ = 0.1. With a tax rate of around 20%, this

implies a consumption loss of around 0.2% of aggregate output in steady state.15 Although

this parameterization is somewhat arbitrary, the focus here is on whether nominal GDP

14 Twenty-year sections were used as a basis for calibration to increase the number of data points available. 15 The steady-state tax rate in the model is 20.4% and the steady state excess burden is 0.21%.

8

targeting raises or lowers the expected tax burden – and this does not hinge crucially on the

calibration of ϕ as shown in the sensitivity analysis section (see Section 4.4).

Other parameters

The private discount factor β is set equal to 0.60, which is equivalent to an annual value of

0.983 if each period lasts 30 years. The coefficient of relative risk aversion, γ, is set equal to

1.5, which is a standard value. The parameter ε is set at 0.30, implying that the share of the

aggregate endowment going to the young is 0.70. Government spending is set at g* = 0.17

and trend inflation at Π* = 1.81, consistent with inflation of 2% per year. The world real

interest rate r* is set at 1.5, which implies a real interest rate of 1.3% per annum. With the

above calibration, the demand for government debt in the deterministic steady state is 0.069.

For the purpose of calibration, the share of indexed government debt, v, was set at 0.25,

which is similar to the current UK share. This share has no impact on deterministic steady-

state but does affect the stochastic steady state and the average tax burden.

2.4.2 Model steady state and key ratios

Table 1 reports the deterministic and stochastic steady states of the model under the baseline

calibration. The model does fairly well in terms of target ratios. The consumption share of

GDP is 0.83 and the government spending share is 0.17. The latter is close to the share in

data, and the former is the implied residual given the absence of investment. The ratio of

government debt to GDP is 0.07, which is similar to the ratio of long-term debt in UK data.16

Finally, the ratio of tax revenue to GDP is 20%, which is the basic rate of UK income tax.17

Table 1 – Model solution versus key ratios (baseline calibration)

Ratio Target Definition Deterministic Stochastic Notes

τ 0.20 Income tax revenue/GDP 0.20 0.20 Target: UK Basic Tax

g/Y 0.17 Govt. spending/GDP 0.17 0.17 Target: UK (ONS)

c /Y 0.83* Consumption/GDP 0.83 0.83 *Investment absent

b /Y 0.08 Long-term govt. debt/GDP 0.07 0.07 Target: UK (ONS, DMO)

3 Results

The model was solved using a second-order perturbation in Dynare (Adjemian et al., 2011) to

obtain approximate theoretical moments. The debt share that minimises the average excess

tax burden is computed as described in Section 2.3. Figure 1 reports average excess burden

and how it changes as the share of indexed debt is increased. Figure 2 provides further detail

by decomposing the expected excess burden into mean and volatility effects (see Equation

(14)), while Figure 3 highlights the role of key variables that drive the results.

16 The UK debt-to-GDP ratio averaged around one-third from 2000 to 2011 (ONS, 2011). With a share of long-

term government debt of 25%, this implies a target long-term government debt to GDP ratio of around 8%. The

Debt Management Office (DMO) classifies gilts as ‘long-term’ if maturity exceeds 15 years; see DMO website. 17 An alternative target would be the UK share of receipts from income and wealth taxes. This share averaged

around 15% of GDP from 2000 to 2012 (HM Treasury, 2013).

9

Fig 1 – Average excess burden and indexation. Figure plots E[f(τ)] in percent of aggregate output as the share

of indexed debt, v, is varied from 0 to 1 (i.e. 100%).

Fig 2 – Decomposition of average excess burden of taxes into mean and volatility effect.

Figure plots the mean effect, [f(E[τ])], and the volatility effect, fττvar[τ], in percent of aggregate output as the

share of indexed debt is varied from 0 to 1. The total equals the average excess burden plotted in Fig 1.

Fig 3 – Real variables affecting the average excess burden. Figure plots the unconditional moments of

variables that matter for the average excess burden through the mean and volatility effects.

10

3.1 Inflation targeting

Under inflation targeting, the average excess burden of taxes falls as the share of indexed

debt is increased (see Figure 1). This is a result of both the mean and volatility components of

the excess burden falling, as shown in Figure 2. Mean taxes fall as the share of indexed debt

is increased because the inflation risk premium is positive (see Figure 3),18 so that issuance of

nominal debt requires higher average taxes in order to maintain government spending. On the

other hand, the volatility of taxes falls as the share of indexed debt is increased because the

volatility of the real return on nominal debt is higher than that on indexed debt (see Figure 3),

because the latter is not subject to inflation risk. This makes the real debt burden more stable

for a given level of past debt, so that less variation in the tax rate is necessary. This also has

the knock-on effect of making expected lifetime consumption growth more stable, so that the

demand for government debt is less volatile as indexation is increased (see Figure 3). This

effect stabilises the real debt burden – and hence the tax rate – even further.

The average excess burden falls by around 0.01% of aggregate output as the economy moves

from only nominal debt to issuing only indexed debt; see Figure 1. It can be seen from Figure

2 that although most of the average excess burden (more than 0.2%) is due to the mean level

of taxes, the reduction in excess burden as the indexation share is increased is driven by a

reduction in tax volatility. In other words, the tax volatility effect is the dominant factor

shaping the relationship between excess burden and the share of indexed debt.

3.2 Nominal GDP targeting

Under nominal GDP targeting, the average excess burden initially falls as the share of

indexed debt is increased, but this is quickly reversed. The excess burden rises sharply after

the minimum point, which occurs at an indexation share of 14% (see Figure 1). Hence, most

government debt should be nominal in order to minimise the excess burden of taxes. This

contrasts with inflation targeting, where the excess burden is minimised by issuing only

indexed debt. This difference in results can be understood in terms of the mean and volatility

effects (see Figure 2). Under nominal GDP targeting mean taxes fall as the share of indexed

debt is increased because the government avoids paying the inflation risk premium when it

issues indexed debt (see Figure 3).19 However, this small effect is dominated by the volatility

effect, which falls until an indexation share of 14% is reached before rising sharply (see Figs

2 and 3). As a result, the overall excess burden rises as the share of indexed debt is increased.

The volatility of taxes initially falls due to a diversification effect: the (output-adjusted) real

returns on indexed and nominal debt are only moderately positively correlated (see Figure 3).

After the minimum, the volatility of taxes rises sharply as the share of indexed debt is

increased because the effect of nominal GDP targeting on stabilising expected marginal

utilities is lessened (see Section 2.3). Intuitively, since this stabilisation effect works through

the effect of countercyclical changes in inflation on the real value of the stock of nominal

government debt, lowering the stock of nominal debt progressively lessens the importance of

this channel. The average excess burden rises by around 0.01% of aggregate output as the

economy moves from only issuing only nominal debt to only indexed debt. Although the

minimum excess burden is achieved at an indexation share of 14%, the numerical value of the

18 Notice that consistent with Equation (10a), the inflation risk premium under inflation targeting falls as the

share of indexed debt is increased and is zero when v = 1. 19 Consistent with Equation (10b), the inflation risk premium under nominal GDP targeting falls as the share of

indexed debt is increased but is positive when v =1.

11

excess burden at this point is almost indistinguishable from that in a nominal debt-only

economy. Hence, issuing only nominal debt provides a fairly good approximation to excess

burden-minimising debt policy.

Comparing inflation and nominal GDP targeting, we see that nominal GDP targeting lowers

average excess burden relative to inflation targeting. Indeed, under the assumption that the

share of indexed debt is set to minimise average excess burden under both regimes, the

excess burden is reduced by 0.011% of aggregate output. Figure 2 makes clear that the

reduction in excess burden under nominal GDP targeting is the result of fall in the volatility

effect (i.e. more stable taxes) which dominates a mild increase in the level effect from higher

mean taxes.20 Figure 3 shows confirms that there is a large proportional reduction in the

variance of taxes. It is worth noting that inflation targeting and nominal GDP targeting have

identical implications in the case where only indexed debt is issued (i.e. v = 1). This is

because monetary policy only matters through the effect of unanticipated inflation on the real

value of government debt – a channel which is ‘shut down’ if all debt is indexed.

3.3 Discussion and policy implications

The above results stress that the share of indexed government debt matters for the excess

burden of taxes and that this relationship is somewhat different under inflation and nominal

GDP targeting. Moving from zero indexation to full indexation implies a reduction in excess

burden under inflation targeting but an increase under nominal GDP targeting, and the

indexation shares that minimise the excess burden are very different under the two regimes.

While inflation targeting favours issuance of only indexed government debt, nominal GDP

targeting calls for issuance of nominal debt with only a small fraction of indexed debt.

It is also interesting that monetary policy has substantial effects on tax volatility: when the

share of indexed debt is set to minimise excess burden, the variance of taxes reduced by

around one-third under nominal GDP targeting (see Figure 3). To the extent that tax volatility

channel drives the difference in results under inflation and nominal GDP targeting, this

channel is likely to be important when comparing social welfare these regimes. Overall, the

results suggest that a change in monetary policy from inflation to nominal GDP targeting

would lower the excess tax burden and be accompanied by a fall in the share of indexed debt.

4 Extensions and robustness

4.1 Closed economy case

The baseline model considers a small open economy. Consequently, the world real interest

rate is taken as given. Since this assumption is not appropriate for all economies, it is

important to investigate whether the main results remain intact when this assumption is

relaxed. We therefore consider a version of the model in which the real return on indexed

debt is determined endogenously as the result of bond market equilibrium. To do so, it is

assumed that the total supply of bonds is held constant at the value b* > 0 that implies the

same steady-state real interest rate as in the baseline case. The results are shown in Figs 4-6.

20 As discussed in Section 2.3, average taxes are higher under nominal GDP targeting because the government

pays a higher inflation risk premium on nominal debt, whereas tax volatility is lower for two different reasons.

12

Fig 4 – Average excess burden and indexation (closed economy). Figure plots E[f(τ)] in percent of aggregate

output as the share of indexed debt, v, is varied from 0 to 1 (i.e. 100%).

Fig 5 – Decomposition of average excess burden of taxes into mean and volatility effect (closed economy).

Figure plots the mean effect, [f(E[τ])], and the volatility effect, fττvar[τ], in percent of aggregate output as the

share of indexed debt is varied from 0 to 1. The total equals the average excess burden plotted in Fig 4.

Fig 6 – Real variables affecting the average excess burden. Figure plots the unconditional moments of

variables that matter for the average excess burden through the mean and volatility effects.

13

The main conclusions remain intact. Nominal GDP targeting lowers the average excess

burden relative to inflation targeting, with the gap closing as the share of indexed debt is

increased. An indexation share of 100% (0%) minimises the excess burden under inflation

targeting (nominal GDP targeting). Hence, the closed economy case reinforces the conclusion

that nominal debt is favoured under nominal GDP targeting. Interestingly, both the mean tax

rate and tax volatility are now lower under nominal GDP targeting (see Figure 6), so that the

volatility effect is reinforced rather than offset by the level effect.

This difference in the results is driven by the fact that the endogenous real return on indexed

debt is lower under nominal GDP targeting because there is a larger precautionary savings

motive as a result of the greater level of uncertainty about consumption in retirement when

inflation risk is increased (see Equation (8)). In turn, the fall in the real interest rate is

sufficient to offset the increase in the inflation risk premium, so that the average real return

on nominal debt falls, and it becomes cheaper to issue both indexed and nominal government

debt under nominal GDP targeting. These results can be understood using the equation:

]ˆ[var5.0ˆ]ˆ[ 1,,1 ttftntt IRPrrE (16)

where

PSE

tttttft cccEr ]ˆ[var5.0]ˆˆ[ˆ1,2

2

,11,2, and PSE is the precautionary savings effect.

As can be seen from Equation (16), an increase in the precautionary savings effect of

sufficient magnitude will the lower the expected real return payable on nominal debt even if

the inflation risk premium rises. This is what happens under nominal GDP targeting. As a

result, average borrowing costs fall despite the increase in the inflation risk premium. Thus,

while the overall impact of nominal GDP targeting on excess burden is unchanged, its effect

on government borrowing costs depends crucially on whether the economy is open or closed.

4.2 Model with money

In the baseline model, money is absent and monetary policy determines the price level

directly. To investigate robustness, versions of the model were simulated in which money is

introduced via money in the utility function as in McCallum (1988) and a cash-in-advance

constraint as in Artus (1995).21 The parameters that determine money demand were set so that

real money balances were around 3% of aggregate output (as in UK data). The other model

parameters were unchanged relative to the baseline case.

Table 2 – Excess burden in the models with money

Model Burden minimising

indexation shares (%)

Minimum excess burden

(% of aggregate output)

MIUF IT: 100 NIT: 0 IT: 0.203 NIT: 0.199

CIA IT: 100 NIT: 0 IT: 0.250 NIT: 0.232

Notes: MIUF = money in the utility function; CIA = cash-in-advance.

Table 2 reports the debt shares that minimise the average excess burden. As in the baseline

model, inflation targeting calls for only indexed debt in order to minimise the excess burden,

in contrast to nominal GDP targeting. Moreover, nominal GDP targeting lowers the average

21 See Crettez et al. (1999) for a review of cash-in-advance constraints in overlapping generations models.

14

excess burden relative to inflation targeting except at indexation shares very close to 1.

Hence, the main conclusions are quite robust to the introduction of money. The full models

and results are reported in the Supplementary Appendix.

4.3 Model with foreign bonds

The baseline model assumes that the government meets household demand for saving

through bond issuance. The Supplementary Appendix reports results for a version of the

model in which this assumption is relaxed, so that foreign debt is held by domestic

consumers. The result that the excess burden is lower under nominal GDP targeting remains

intact, and this is still driven by lower tax volatility. The share of indexed debt that minimises

excess burden is 100% under inflation targeting and 71% under nominal GDP targeting.

4.4 Parameter sensitivity analysis

To investigate robustness, each of the model parameters was assigned a ‘high’ and ‘low’

value and the results were recalculated.22 The main conclusions are not sensitive to

calibration. In particular, the average excess burden of taxes remains lower under nominal

GDP targeting, and this result is driven by the tax volatility effect dominating the tax level

effect. In addition to this, the share of indexed debt that minimises the excess burden remains

at 100% under inflation targeting but is less 50% under nominal GDP targeting. The full

results are reported in the Supplementary Appendix.

5 Conclusion

This paper has investigated the impact of nominal GDP targeting on the excess burden of

distortionary taxes. The analysis is motivated by the fact that nominal GDP targeting has two

opposing effects on the excess burden: on the one hand, the average level of taxes is higher;

on the other, taxes are less volatile. Numerical analysis is needed to determine which effect

dominates. The main finding is that, under plausible calibrations, the average excess burden

is lower under nominal GDP targeting because tax volatility is somewhat lower than under an

inflation targeting regime. In addition, while issuing only indexed government debt

minimises the excess tax burden under inflation targeting, nominal GDP targeting requires

mostly nominal debt in order to do this. The above results appear to be rather robust within

the overlapping generations model studied here. The finding that nominal GDP targeting

would reduce the tax burden relative to inflation targeting highlights an important channel

through which this regime could lead to improvements in social welfare. More generally, the

results show that the choice of monetary regime can have non-trivial implications for the

excess tax burden, and that the magnitude of these effects will depend upon government debt

policy through the share of indexed debt.

22 Results are not reported for the trend inflation parameter Π* as there was no change in results in this case.

15

References

Adam, K. and Zhu, J. 2015. Price level changes and the redistribution of nominal wealth across the Euro Area.

ECB Working Paper no. 1853.

Adjemian, S., Bastani, H., Juillard, M., Karamé, F. Mihoubi, F., Perendia, G., Pfeifer, J., Ratto, M., and

Villemot, S. 2011. Dynare: Reference Manual, Version 4. Dynare Working Paper No. 1, CEPREMAP.

Artus, P. 1995. Macro Economie (collection Economica Poche). Paris, France: Economica.

Bean, C.R. 1983. Targeting nominal income: an appraisal. Economic Journal 93(272), 806-819.

Bekaert, G., Wang, X. 2010. Inflation risk and inflation risk premium. Economic Policy 25, 755-806.

Billi, R.M. 2013. A note on nominal GDP targeting and the zero lower bound. Sveriges Riksbank Working

Paper No. 270 (Revised May 2015).

Bradley, M.D., Jansen, D.W. 1989. The optimality of nominal income targeting when wages are indexed to

price. Southern Economic Journal 56(1), 13-23.

Campbell, J.Y., Shiller, R.J., Viceira, L.M. 2009. Understanding inflation-indexed bond markets. Brookings

Papers on Economic Activity 40(1), 79–120.

Champ, B., Freeman, S. 1990. Money, output, and the nominal national debt. American Economic Review 80,

390-397.

Crettez, B., Michel, P. and Wigniolle, B. 1999. Cash-in-advance constraints in the Diamond overlapping

generations model: neutrality and optimality of monetary policies. Oxford Economic Papers 51(3), pp. 431-52.

Department of Finance Canada 2011. Debt Management Report 2010-2011. Available at:

http://www.fin.gc.ca/dtman/2010-2011/dmr-rgd11-eng.asp. Last accessed: Nov 9, 2015.

Doepke, M., Schneider, M. 2006. Inflation and the redistribution of nominal wealth. Journal of Political

Economy 114(6), 1069-97.

Doepke, M., Schneider, M. 2005. Redistribution of nominal wealth and the welfare cost of inflation.

Unpublished working paper.

Hall, R.E. and Mankiw, N.G. 1993. Nominal income targeting. NBER Working Paper No. 4439.

HM Treasury 2013. Budget 2013. Available at: https://www.gov.uk/government/publications/budget-2013-

documents. Last accessed: Dec 11, 2015.

Koenig, E.F. 2013. Like a good neighbour: monetary policy, financial stability, and the distribution of risk.

International Journal of Central Banking 9(2), 57-82.

McCallum, B.T. 1988. The optimal inflation rate in an overlapping-generations economy with land. NBER

Working Paper No. 1892.

Meh, C.A., Rios-Rull, V.J., Terajima, Y. 2010. Aggregate and welfare effects of redistribution of wealth under

inflation and price-level targeting. Journal of Monetary Economics 57(6), 637-52.

Meh, C.A., Terajima, Y. 2011. Inflation, Nominal Portfolios, and Wealth Redistribution in Canada. Canadian

Journal of Economics 44(4), 1369-1402.

ONS 2011. Financial Statistics (June 2011 Edition). Office for National Statistics, UK.

Sheedy, K.D. 2014. Debt and incomplete financial markets: a case for nominal GDP targeting. Brookings

Papers on Economic Activity 48(1), 301-373.

Woodford, M. 2003. Interest and prices: Foundations of a theory of monetary policy. Princeton University

Press: Princeton, NJ.

16

Supplementary Appendix (For online publication only)

Section A – Models with money

This section introduces money into the model as a robustness check. The amended model is set out

and results are presented for two cases: (i) money in the utility function, (ii) cash-in-advance.

A.1 Model and first-order conditions

Consider a model with money. The stock of money in period t is denoted Mt and follows a fixed rule.

Changes in the money stock are accomplished by lump-sum money transfers to the old, St = Mt – Mt-1,

which are taken as given by households. Since seignorage revenue finances lump-sum subsidies to the

old, the government budget constraint is unchanged. Real money balances are given by mt = Mt/Pt.

The real money transfer received by the old is defined as st = St/Pt.

Money in the utility function

A young agent born at date t solves the following problem:

)()()( max 1,2,1,

d

tmtttbb

mUcUEcUit

nt

s.t. d

t

i

t

n

tttttt mbbYfYc )]([)1(,1 (A1)

1

1

1,111,2 )/(

t

d

tt

i

tft

n

ttttt smbrbRYc

where 11 )()1()( d

t

d

t mmU and δm > 0 is the weight on utility from real money holdings.

The first-order conditions for bonds and real money holdings are:

])([)( 1

11,2,1

ttctttc cUERcU (A2)

)]([ )( 1,2,,1 tctfttc cUErcU (A3)

])([)()( 1

11,2,1

ttct

d

tmmtc cUEmUcU (A4)

The goods and bond market clearing conditions are unchanged, but there is additionally a money

market clearing condition which requires that money supply equal money demand:

d

ttt mPM (A5)

Notice that Equation (A5) implies that inflation is given by

d

t

d

t

tt

t

t

tmm

MM

P

P

1

1

1 /

/

(A6)

It is assumed that the money supply is set so that inflation is given by Equations (6) and (7) from the

main text. That is,

)/( 11

d

t

d

t

j

ttt mmMM where

NIT when )exp(

)exp()/(*

IT when )exp(*

1

1 jYY

j

t

t

tt

t

j

t

(A7)

We set δm = 0.01 so that the ratio of money balances to GDP is around 3% as in UK data. All other

parameter values are the same as in the baseline calibration.

17

Cash-in-advance constraint

Consider a model of fiat money with a simple cash-in-advance constraint as in Artus (1995). Each

young agent is required to hold fiat money worth at least a fraction δ > 0 of their consumption, and

this constraint is assumed to be binding so that real money holdings are mt = δc1,.t for all t.23

A young agent born at date t solves the following problem:

)()( max 1,2,1,

tttbb

cUEcUit

nt

s.t. d

t

i

t

n

tttttt mbbYfYc )]([)1(,1 (A8)

1

1

1,111,2 )/(

t

d

tt

i

tft

n

ttttt smbrbRYc

t

d

t cm ,1

The first-order conditions for bonds and real money holdings are:

tttctttc cUERcU

])([)( 1

11,2,1 (A9)

ttctfttc cUErcU )]([ )( 1,2,,1 (A10)

tttcttc cUEcU )1(])([)( 1

11,2,1

(A11)

Notice that Equations (A9) and (A11) imply that

])([)1( 1

11,21

ttcttt cUER (A12)

The money market clearing conditions is:

tt

d

ttt cPmPM ,1 (A13)

Notice that Equation (A13) implies that inflation is given by

1,1,1

1

1 /

/

tt

tt

t

t

tcc

MM

P

P (A14)

As with money in the utility function, it is assumed that the money supply is set so that inflation is

given by Equations (6) and (7). That is,

)/( 1,1,11 tt

j

ttt ccMM (A15)

where

NIT when )exp(/)exp()/(*

IT when )exp(*

11 jYY

j

tttt

tj

t

.

We set δ = 0.065 so that the ratio of money balances to GDP is around 3% as in UK data. All other

parameter values are the same as in the baseline calibration.

23 The cash-in-advance constraint will be binding as long as Rt > 1 for all t, as shown by Equation (A12) below.

18

A.2 Numerical results

Money in the utility function (δm = 0.01)

Fig A1 – Average excess burden and indexation. Figure plots E[f(τ)] in percent of aggregate output as the

share of indexed debt, v, is varied from 0 to 1 (i.e. 100%). MIU calibration: δm = 0.01.

Fig A2 – Decomposition of the average excess burden of taxes into mean and volatility effect.

Figure plots the mean effect, [f(E[τ])], and the volatility effect, fττvar[τ], in percent of aggregate output as the

share of indexed debt is varied from 0 to 1. MIU calibration: δm = 0.01.

Fig A3 – Real variables affecting the average excess burden. Figure plots the unconditional moments of

variables that matter for the average excess burden of taxes. MIU calibration: δm = 0.01.

19

Cash in advance constraint (δ = 0.065)

Fig A4 – Average excess burden and indexation. Figure plots E[f(τ)] in percent of aggregate output as the

share of indexed debt, v, is varied from 0 to 1 (i.e. 100%). CIA calibration: δ = 0.065.

Fig A5 – Decomposition of the average excess burden of taxes into mean and volatility effect.

Figure plots the mean effect, [f(E[τ])], and the volatility effect, fττvar[τ], in percent of aggregate output as the

share of indexed debt is varied from 0 to 1. CIA calibration: δ = 0.065.

Fig A6 – Real variables affecting the average excess burden. Figure plots the unconditional moments of

variables that matter for the average excess burden of taxes. CIA calibration: δ = 0.065.

20

Section B – Model with foreign bonds

B.1 Model

Foreign bonds, bF, pay the constant world real interest rate rf = r*. Demand for these bonds is

generated by the assumption that the government supplies a fixed amount of domestic debt b* > 0,

which falls short of total demand for saving. Under these circumstances, households will make

purchases of foreign bonds up to the point where this excess demand is eliminated. Household budget

constraints must be amended and there is now an additional household first-order condition.

A young agent born at date t solves the following problem:

)()( max 1,2,1,

tttbbb

cUEcUFt

it

nt

s.t. F

t

i

t

n

tttttt bbbYfYc )]([)1(,1 (B1)

F

tft

i

tft

n

ttttt brbrbRYc ,,111,2 )/(

The first-order conditions for home and foreign bond holdings are:

tttctttc cUERcU ])([)( 1

11,2,1

(B2)

ttctfttc cUErcU )]([ )( 1,2,,1 (B3)

The market clearing condition in the home bond market is

*bbb s

t

d

t t (B4)

where sn

t

si

t

s

t bbb ,, is the total supply of government debt.

Market clearing in goods is now )](1[* 1,1,2,1 tt

F

tft

F

ttt fYbrbgcc .

The numerical results set b* = 0.054 because this amounts to 80% of the total demand for debt in

steady state (so that the other 20% is foreign debt). Other than home bonds, the deterministic steady-

state of the model is identical to the baseline case.

B.2 Numerical results

Fig B1 – Average excess burden and indexation. Figure plots E[f(τ)] in percent of aggregate output as the

share of indexed debt, v, is varied from 0 to 1 (i.e. 100%). Home bond supply: b* = 0.054.

21

Fig B2 – Decomposition of the average excess burden of taxes into mean and volatility effect.

Figure plots the mean effect, [f(E[τ])], and the volatility effect, fττvar[τ], in percent of aggregate output as the

share of indexed debt is varied from 0 to 1. Home bond supply: b* = 0.054.

Fig B3 – Real variables affecting the average excess burden. Figure plots the unconditional moments of

variables that matter for the average excess burden of taxes. Home bond supply: b* = 0.054. The mean and

variance of government debt are omitted from the figure because the home bond supply is constant.

Discussion

The main conclusions remain intact: the average excess burden is lower under nominal GDP targeting

and this result is driven by the tax volatility effect dominating the tax level effect. Excess burden is

minimised at an indexation share of 100% under inflation targeting and 71% under nominal GDP

targeting. The latter figure is somewhat higher than in the baseline model (14%) because the tax

volatility effect is smaller. This is because while government debt is less volatile under nominal GDP

targeting in the baseline model, this effect is absent in the model with foreign bonds because the

supply of home government debt is constant.

22

Section C – Parameter sensitivity analysis

Sensitivity of total excess burden and indexation

Fig C1 – Excess burden and indexation: high calibrated values. Figure plots E[f(τ)] as the share of indexed

debt is varied from 0 to 100%. IT denotes inflation targeting; NIT denotes nominal GDP targeting.

Fig C2 – Excess burden and indexation: low calibrated values. Figure plots E[f(τ)] as the share of indexed

debt is varied from 0 to 100%. IT denotes inflation targeting; NIT denotes nominal GDP targeting.

23

Table C1 – Excess burden minimising indexation shares (in %)

Parameter High calibration case Low calibration case

Output persistence: ρY IT: 100 NIT: 28 IT: 100 NIT: 0

Output volatility: σY IT: 100 NIT: 33 IT: 100 NIT: 0

Control error volatility: σ IT: 100 NIT: 0 IT: 100 NIT: 47

Excess burden parameter: ϕ IT: 100 NIT: 12 IT: 100 NIT: 14

Discount factor: β IT: 100 NIT: 14 IT: 100 NIT: 12

Risk aversion: γ IT: 100 NIT: 14 IT: 100 NIT: 9

Income share of old: ε IT: 100 NIT: 0 IT: 100 NIT: 34

World real interest rate: rf IT: 100 NIT: 14 IT: 100 NIT: 11

Government spending: g* IT: 100 NIT: 0 IT: 100 NIT: 24

Notes: The calibrations are listed Fig C1 and Fig C2. Baseline case: IT: 100%; NIT: 14%.