Debt projections and fiscal sustainability with feedback effects

Abstract This paper analyses long-term fiscal sustainability with a model which incorporates a number of feedback effects. When fiscal policy responds to ensure long-term sustainability, these feedback effects can potentially modify the intended outcomes by either enhancing or dampening the results of the policy interventions. The feedbacks include the effect on labour supply in response to changes in tax rates, changes in the country risk premium in response to higher public debt ratios, and endogenous changes in the rate of productivity growth and savings that respond to interest rates. A model of government revenue, expenditure and public debt which incorporates these feedbacks is used to simulate the outcome of a range of fiscal policy responses. In addition, the effects of population ageing and productivity growth are explored.

there are several aspects of the exercise that could be improved in future LTFSs, and the Treasury should continue to refine its analytical tools to do so. First...the non-behavioural, spreadsheet-based nature of the LTFM implies that projections do not allow for feedbacks from the fiscal developments to the macro-economy. ... it would be desirable to present, in future versions of the LTFS, scenarios with different dynamic paths of the key macroeconomic assumptions, to allow for plausible feedbacks from the growth of the debt.
The present paper, therefore, represents a first step in responding to this challenge. It examines long-term fiscal sustainability in the context of a modified 'bottom-up' model in which a limited number of feedback effects are introduced, while a mechanical approach continues to be used for many components of the model. Policy responses to fiscal deficits, along with other endogenous responses to debt levels, have feedback effects. These may enhance or modify the intended or initial consequences of those responses. For example, tax and expenditure policy changes might be implemented to deal with a fiscal deficit. At the same time, the interest rate may vary as a result of risk premium adjustments to debt levels. This may in turn have further consequences for fiscal sustainability.
However, rather than attempting to capture all the details involved in many types of expenditure and tax, the present paper uses a more aggregative approach than the Treasury's LTFM. It distinguishes only four types of expenditure in addition to debt servicing costs, and has a very simple income tax structure together with a Goods and Services Tax (GST); other forms of tax revenue are combined into a single component. The feedback effects are modelled using reduced-form specifications rather than a structural approach with explicit optimising behaviour. The model, nevertheless, contains a sufficient amount of detail to enable a range of policy responses to be examined. Furthermore, careful calibration of the model produces a 'benchmark' projection of the ratio of government debt to income that closely approximates that of the Treasury's LTFM (2013c). The basic structure presented here has also been influenced by the desire in future research to introduce uncertainty into the model and to examine optimal policies to achieve sustainability, requiring a specified evaluation function.
Faced with a set of revenue and expenditure projections implying an increase in debt over a specified time period, a range of fiscal sustainability or solvency indicators can be produced, based on manipulations of the government multi-period budget constraint. The many issues involved in assessing sustainability and the required adjustments in the face of projected debt growth are discussed by Buckle and Cruickshank (2014) in the New Zealand context. Early definitions and measures were proposed by Blanchard, Chouraqui, Hagemann, and Sartor (1990). The approach adopted by the European Union is set out in detail in European Commission (2006) and for an example of its use, see Kleen and Pettersson (2012). For a non-technical discussion of issues, see Schick (2005), and a review of alternative approaches is by Pradelli (2012).
Basic measures include the increase in the fiscal balance (the difference between revenue and expenditure including debt interest charges) in each year, expressed as a proportion of GDP, needed for the present value over an infinite horizon of surpluses to cover the current debt. Alternatively, a less restrictive measure is the increase in the fiscal balance (again as a proportion of GDP) needed to attain a specified debt target by the end of the projection period. In European Commission (2006), the first measure is denoted by S2, and the second, involving a debt ratio of 60% of GDP by 2050, is referred to as S1 (see Appendix 2 for further details of these measures).
These approaches are acknowledged to provide only an indication of the risk facing a country, and do not pretend to offer an optimal response. In addition, the measures ignore the time path of debt, since they relate to a required constant (relative) increase in the fiscal balance each period. The time profile may itself have consequences which raise important policy concerns. Furthermore, no consideration is given to how attainable the alternative objectives may be, and which policy instruments might be used. By contrast, the present paper considers explicit policy variations needed to achieve a specified fiscal balance at the end of the projection period.
The basic model is set out in Section 2. Feedback and other endogenous effects are added in Section 3. Benchmark calibration values are described in Section 4. Benchmark projections are presented in Section 5, where it is shown that, in the absence of feedback effects, where expenditure items are assumed to grow at specified fixed rates and tax rates are unchanged over time, the model can closely approximate the projections obtained by the Treasury's LTFM.
Having described the model, policy simulations are reported in Section 6. In the benchmark simulations, the main difference between the model with and without feedback effects arises as a result of the rising risk premium, and hence debt servicing costs, as the debt ratio increases. However, unlike a number of other countries, the debt ratio in New Zealand is not projected to increase to the levels that generate very large increases in the risk premium. Other feedbacks are largely absent in the benchmark case because growth rates are held constant and there are no tax policy changes. Of interest are cases where expenditure and tax policy changes are imposed with particular objectives in mind. For example, if the income tax or indirect tax rates are increased, or various expenditure growth rates are reduced in an attempt to control the extent of the debt increase, other feedback effects play a more significant role. Conclusions are in Section 7.

The basic model
This section provides a description of the main components of the model. As explained in the introduction, the aim is to construct a model that is capable of projecting the paths of government revenue and expenditure, and therefore debt, under a range of assumptions and feedback effects. To make the model as transparent as possible, a high level of aggregation is used. It is clearly necessary to allow demographic variations in both population size and its age composition to influence government expenditure and revenue. While detailed demographic projections are used, distinctions are drawn only between those of working age, retirement age and those below working age.

Government expenditure and debt
Given that a primary concern is with fiscal sustainability and with policies designed to achieve sustainability, the evolution of government debt plays a crucial role in the model. Let D t denote debt at the end of time period, t, for t D 1, …, T, where D 0 is the debt inherited from the past and D Ã T is the target debt level for the end of the planning period, T. If r t is the domestic interest rate at time t, equivalent to the government bond rate, then the debt servicing cost at time t, denoted d t , is given by (1) The interest rate depends on the world interest rate, r w , which is assumed to be constant (the small country assumption), and a risk premium, r p, t . Thus, In addition to debt servicing costs, government expenditure includes welfare spending, W t , which consists of two components. They are transfer payments (welfare benefits) of W B,t , received by nonpensioners, and superannuation benefits of W S,t received by pensioners. New Zealand Superannuation (NZS) is taxable, as are most of the working-age transfer payments, and is universal. This is allowed for in the calibration of the model, discussed below, which uses net-of-tax values. Hence The levels per person are denoted as W Ã S, t and W Ã B, t , so that if N S,t and N B, t denote the number in receipt of the pension and welfare benefits, respectively, All other spending at t is denoted by E t . This is composed of spending on publicly provided goods such as health and education, E I,t , and other expenditure, E O,t , which includes, for example, core government services, law and order, and defence: hence E t D E I, t C E O, t . The former may be considered as investment in human capital, while the other expenditure has no direct impact on individuals. As explained below, E O, t is assumed to have no direct impact on the labour supply, and thus incomes, of individuals. While E I, t does not have a direct impact, it influences income via its effect on productivity growth. Variations in these spending categories are produced by variations in per capita amounts, E Ã I, t and E Ã O, t and variations in the total population, N t : Total government expenditure, G t , is thus Define R t as total tax revenue from direct and indirect taxes: this is considered in more detail below. The debt in t is thus given by Substituting (4) into (5) gives Continual substitution gives the long-term government budget constraint as The simpler form of this budget constraint, for the case where the rate of interest is constant, is used in Appendix 2 to examine the annual increase in the fiscal balance, R t ¡ G t , as a ratio of GDP, needed to achieve a target debt ratio by a given year.

Income generation
For the calculation of tax revenues, it is necessary to obtain the time profile of aggregate income, denoted by Y A,t at time t. This is the sum of incomes arising from labour and (capital) rental income, Y t , and interest income from financial savings. The model makes no attempt to treat the production side of the economy explicitly. The model thus contains no explicit wage rate, nor does it deal with labour and capital inputs into production. The high level of aggregation also means that the model cannot deal with a changing composition of output and any relative price changes which may result from population ageing and government policy. A base level of productivity is taken as exogenously given and, as explained below, productivity changes can arise from growth in public expenditure on health and education per person, which is considered to augment human capital. First, define Y P,t as total 'potential income' in period t. To allow for productivity growth at the rate r t , write Let L t indicate the ratio of actual to potential income, so that aggregate income can be written as Hence L t captures all possible incentive effects. The specification of L t is described in Section 3.3. Interest income then needs to be added. Assume that all forms of income are taxed at the same rate. Then if S t denotes aggregate financial savings at time, t, as defined above, these are all assumed to be invested at the going rate, r t . Letting financial capital be denoted by K t , then As this refers to the accumulation of financial savings, no depreciation is applied. As discussed above, the production side of the economy, including investment and capital accumulation, is not modelled explicitly. Hence aggregate income is For simplicity, this assumes that the borrowing and lending rates are equal, and the same both for the government and individuals, and the return to investment is equal to the domestic rate of interest.
The above specification can easily be augmented to allow for population growth. A simple adjustment is made by raising Y A,t by a proportion that depends on the growth rate, from period t ¡ 1 to t, of the population above working age.

Tax revenue
No attempt is made here to model the complexity of the tax and transfer system. Suppose that income tax is simply a constant proportion, t t , of taxable income. Income tax revenue is thus easily obtained as t t Y A, t . Tax revenue is also obtained from indirect taxes. Define V t as indirect tax revenue at t, from a GST/VAT type of system, where v t is the tax-exclusive rate applied to all expenditure. However, indirect taxes applied to E t are ignored here since these are netted out in the government's budget constraint. The tax-inclusive indirect tax rate is v t /(1 C v t ).
First, it is necessary to obtain expenditure, inclusive of indirect tax. Savings, S t , are made from net income. Assume that all transfer payments, W t , are consumed. Then if savings are a constant proportion, s t , of post-tax income, Indirect tax is thus Total tax revenue, R t , consists of income tax, plus V t , plus other revenue, R O,t . The latter is specified as an amount per capita, R Ã O,t , which is subject to an exogenous growth rate, along with growth arising from the increase each period in the population above working age. In considering the second term in (13), W/(1 C v) can be regarded as the tax-exclusive value of expenditure, on which the taxexclusive rate, v, is levied.
Total revenue is thus Substituting for V t from (13) gives total revenue as where t Ã t is the overall effective proportional income tax rate, given by The term, (1 ¡ s t )(1 ¡ t t )/(1 C v t ), reflects the tax-exclusive expenditure arising from an extra dollar of gross income from labour and capital. This is subject to indirect tax at the tax-exclusive rate, v t . Hence t Ã t reflects the combined effect of the income and consumption tax rates.

Feedback effects
This section describes feedback effects involving the risk premium, savings, incentives and productivity growth. In each case, simple reduced-form specifications are adopted rather than attempting to introduce microfoundations into the model. Given the absence of an explicit production function, the wage rate is not endogenous and, with only aggregate output modelled, there are no relative price effects. The model does not have an explicit role for the exchange rate (which also affects relative prices), and its possible connections with the debt ratio and the interest rate risk premium. One possible extension may be to distinguish between traded and non-traded goods, which have different capital intensities. Government expenditure may be considered to be mainly on nontraded goods. For a model using this distinction, see Guest and Makin (2013). In addition, there is no mechanism for the real interest rate to influence investment and, via this effect, the growth rate. Furthermore, investment affects capital intensity and thus wage rates, which in turn affect labour supply. This potential feedback is thus excluded from the present model. Taxfinanced government expenditure has no direct stimulus effect on the real economy except that, as discussed below, the expenditure on health and education is treated as affecting human capital and thus productivity.
The model thus contains only a limited number of possible feedbacks, given the aim of taking an initial step towards introducing endogeneities and linking policy responses to particular policy instruments. Furthermore, the model provide the basis for possible extensions, in particular the introduction of uncertainties and the investigation of optimal policies.

The risk premium
Interest rates in New Zealand typically appear above those in comparator countries. This differential is widely attributed to the presence of a risk premium. Foreign investors in securities denominated in New Zealand dollars demand a margin above the world rate. Burnside (2013) attributes this compensation to the possibility of a depreciation of the New Zealand dollar following a rare and extreme event. The higher is the ratio of public debt to GDP, the more vulnerable the New Zealand economy is to some unexpected event and the greater the risk of a devaluation. Baldacci and Kumar (2010), using a panel of 31 countries for the years 1980À2008, find that 'higher fiscal deficits and public debt raise long-term nominal bond yields in both advanced and emerging markets' (2010, p. 13).
They report that typically 'an increase in the debt ratio of 1 percentage point of GDP leads to an increase in bond yields of around 5 basis points'. In an analysis of an extreme event, Gereben, Woolford, and Black (2003, p. 3) estimate that an outbreak of foot and mouth disease could raise the net public debt by approximately 10 percentage points after five years, with an associated '50 basis point increase in the risk premium on New Zealand dollar assets, as a result of foreign investors becoming more reluctant to invest in New Zealand in times of high uncertainty'.
A number of studies have made estimates for New Zealand. Hawkesby, Smith, and Tether (2000) examine the interest rate differentials between New Zealand and Australia and the United States. They decompose the differentials into expected currency movements, default and liquidity risks, and unexpected currency movements. They estimate that the 10-year currency risk premium is between 1 and 2 percentage points relative to the USA. For the present model, it is assumed that the risk premium at time t is a function of D t¡1 /Y A, t¡1 D DR t¡1 . Ostry, Ghosh, Kim, and Qureshi (2010) show how the cost of borrowing typically rises with higher debt levels. However, their evidence suggests that the risk premium increases only slowly for relatively small values of this ratio, but increases rapidly once it exceeds about 1.5. The response of the risk premium to debt ratios in New Zealand is also discussed by Fookes (2011, p. 11) in the context of a scenario analysis of shocks to New Zealand's fiscal position.
A specification that can capture this kind of relationship is the following. For DR t¡1 in excess of a threshold value, denoted DR Ã , suppose and for DR t ¡ 1 DR Ã , the premium increases linearly: This specification is used to ensure that there is no discontinuity between the two segments. The response to increasing debt ratios, therefore, produces a rise in the risk premium, which has a further consequence for debt as a result of the higher interest cost involved in servicing the debt. Hence this type of endogeneity has important consequences for the evolution of debt. However, there are additional consequences as a result of the influence, directly and indirectly, of changes in the interest rate.

The saving rate
A further possibility is to suppose that the saving rate, s t , depends on the interest rate. However, in principle, this effect is ambiguous, as it comprises both an income and a substitution effect, with opposing influences. In the simulations reported below, it is assumed (in the 'benchmark case') that the interest-elasticity of savings is small but positive. This is reflected in a reduced-form relationship between s t and r t , with ds t /dr t > 0. For simplicity, suppose where parameters u 11 and u 12 are both positive. With a fixed world interest rate of r w , the domestic rate, r t , varies according to the risk premium, r p,t , which depends on the debt ratio, as discussed above. A higher debt ratio may also lead to a Ricardian adjustment in the form of increased savings, if the higher debt were to create expectations of higher future tax rates; but this is not modelled explicitly here. For a review of Ricardian equivalence, see Seater (1993). Similarly, the model does not allow for a possible effect on savings of changes in government expenditure (particularly adjustments to the growth of superannuation and other welfare spending per person).
An increasing debt ratio, therefore, not only leads to a rise in the interest rate, which increases debt repayment costs, but also to a direct effect on the savings rate. For a given change in aggregate income, this mitigates slightly the direct effect of debt on the interest rate, since it is the debt measured in excess of savings, D t¡1 ¡ S t¡1 , that enters into the determination of the risk premium. Furthermore, the savings rate enters into the determination of the effective tax rate, t Ã t , as shown in (16). A higher savings rate reduces the effective tax rate, thereby reducing revenue in the relevant period. This revenue-reducing effect, therefore, slightly reinforces the increase in debt over time. The future tax payments arising from any dissaving is ignored here. It is the aggregate saving rate which varies over time, not the rate in a life-cycle framework.

Incentive effects
It is important to allow for incentive effects of taxation. 2 Suppose the ratio of actual to potential income, L t , is a function of the tax rate, so that L t D L(t Ã t ), with dL t /dt Ã t < 0. As explained above, this function reflects the extent to which combined employment and rental income deviates from its potential. Hence incentive effects here are broader than simply labour supply effects. It is, therefore, useful to borrow from the extensive literature on the elasticity of taxable income, where this concept is defined with respect to the effective net-of-tax rate, 1 ¡ t Ã t . The following constant-elasticity form is ubiquitous: This combines a range of adjustments to taxable income in a simple reduced-form expression. This assumes there are no income effects and the elasticity of L with respect to the net-of-tax rate, 1¡t Ã t , is constant at u 9 . For extensive references to the literature, and estimates for New Zealand, see Carey, Creedy, Gemmell, and Teng (2015).
When the debt ratio is increasing, the endogeneity of both the risk premium and the savings rate means that taxable income is somewhat higher than otherwise because the effective tax rate falls. There is thus a 'tax rate' effect and two 'tax base' effects, moving in opposite directions. A higher debt ratio leads to a higher rate of interest, which raises the savings rate, leading to a fall in the tax base (via the effect on GST). In addition, there is also a fall in the effective tax rate, leading to a rise in the tax base via the effect on work incentives.

Productivity
Investments in the quality of human capital through both health and education can enhance labour productivity. The Treasury (2013b, p. 21) suggests that 'increasing levels of qualifications should have a positive impact on labour market productivity'. Earle (2010, p. 1) argues that, for New Zealand, 'there is evidence that increases in tertiary education have contributed to productivity growth'. This is reinforced by the work of Razzak and Timmins (2010) who found that university qualifications had a positive effect on average economy-wide productivity. 3 Similarly, there is evidence that health affects productivity through various channels. Bloom, Canning, and Sevilla (2001) found that good health has a positive, sizeable and statistically significant effect on economic growth. Bloom and Canning (2003) treat health as part of human capital and assess its impact on economic performance. In subsequent work, Bloom and Canning (2005) find that for developing economies a one percentage point increase in adult survival rates increases labour productivity by about 2.8%.
In a wide-ranging review of possible productivity effects of population ageing, Guest (2014, p. 165) concluded that it 'could affect productivity through a number of mechanisms. But the magnitude and even direction of some of these effects are unclear in theory and evidence'. Infrastructure spending, not considered separately here, may also be growth enhancing.
Suppose that changes in r depends on previous growth of the per capita public expenditure component, E Ã I, t , since this includes education and health expenditure. The change in r depends on the change ' years previously, that is in E Ã I, t¡' . This is assumed to be subject to decreasing returns. Hence if a dot above a variable indicates a proportionate change, with, for example, This logistic form captures decreasing returns, such that the change in productivity growth is a decreasing function of the change in public expenditure. 4 Hence, if r B is a 'base level' of productivity change, If E Ã I, t grows at a constant rate over time, so that _ E Ã I;tÀ' is constant for all t, productivity growth remains constant. A response to the anticipated debt increases which involves cutting the rate of growth of per capita expenditure on health and education therefore has the effect of slowing down the growth of incomes somewhat. Hence tax revenue would be lower than without this feedback effect. It is not necessary here to consider all determinants of productivity, only the potential influence of relevant variables contained within the model. Other influences would included, for example, international connectedness and knowledge-based capital. Since the production side is not modelled here, productivity growth can be regarded as total factor productivity growth, or either labour or capital augmenting.

Calibration of the model
The first step in using the model is to specify time profiles for the expenditure components, E and W, along with starting values for the various revenue and debt variables. Despite the 'simplicity' of the model, suitable orders of magnitude of many of the variables can be obtained from National Income data and demographic projections. The data sources and values are set out in detail in Appendix 1. As shown below, these parameters produce baseline debt ratio projections that are in line with independent results using the Treasury's LTFM.
Parameter values used for the various functions are listed in Table 1. The parameters of the savings function, u 11 and u 12 , were set to ensure that the elasticity of s with respect to r is 0.1 at a rate of Risk premium: u 8 1.0 u 9 0.5 Saving rate:s t D u 11 C u 12 r t u 11 0.03 u 12 0.0833 0.04. The results reported below are obtained using these benchmark parameters. In addition, limited sensitivity analyses are reported in Appendix 3. Figure 1 illustrates the implications for the risk premium of the benchmark calibration values. This form is consistent with the results reported by Baldacci and Kumar (2010). An increasing debt ratio produces modest steady increases in the risk premium until the debt ratio exceeds 100% of GDP (since DR Ã D 1), after which it increases more rapidly, following the quadratic segment of the profile. An increase in the debt ratio from 0.5 to 1.0 causes the risk premium to rise by 0.00075, whereas an increase from 1.0 to 1.5 leads to a rise in the premium of 0.003. A further increase from 1.5 to 2.0 involves a much larger increase in the debt premium of 0.01125.

A benchmark simulation
This section reports benchmark projections, where it is assumed that there are no changes in tax rates and all expenditure categories (per capita) grow at constant rates over the period, using the initial values and parameters described in the previous section. This is the typical 'no change' assumption used in producing expenditure and debt projections. Obviously, such projections of an unsustainable debt ratio path are not regarded in any sense as forecasts, but merely highlight the need for some kind of adjustment. Furthermore, there may be market adjustments (operating for example via wage and price effects) which modify the debt increase. In addition, the partial approach used does not allow for the potential adjustments arising from associated current account positions and exchange rate movements.
The results are shown in Figure 2. Here, the dashed line indicates the debt ratio in each year on the assumption that there are no feedback or endogenous effects. The figure also shows the base projections obtained by the Treasury's LTFM. The solid line shows the projections allowing for the various feedbacks, implying slightly higher debt ratios in the later years. The relatively small role played by feedback effects in this benchmark case should not be interpreted as implying that feedbacks are in general unimportant. The various tax and growth rates are held constant over the period in the benchmark case, so that the only relevant feedback effect arises from the effect on the risk premium of the increasing debt ratio. This increase in the risk premium is, by assumption, quite modest over the range of debt ratios generated by the projections. If the projection period were extended, the debt ratio, by moving above 150% of GDP, would clearly move into the range where a rapid rise in the risk premium, and thus in debt service charges, is generated. Hence the difference between the no-feedback and feedback cases would be much larger. 5 The projections demonstrate a potentially unsustainable situation were there to be no adjustments to the fiscal balance via taxation or revenue changes. The following section considers a number of policies designed to generate sustainable debt ratios. However, it is first useful to consider the separate contribution of population ageing to the debt ratio projections, given much of the focus of the public debate on the demographic transition. Figure 3 compares the benchmark debt ratio projections and those obtained under the assumption that the population age structure remains fixed at its 2014 values, while still allowing the total population to grow at the same rate as in the benchmark projections. Clearly the lack of long-term sustainability arises primarily from demographic changes rather than fundamental problems with tax and expenditure design settings.
The limited feedback effects modelled here clearly do not lead to adjustments which could modify the population ageing effects. With an assumption (common to all projection models) of constant growth rates of expenditure, there is a consequent constant growth rate of income: higher growth via productivity gains requires a change in the growth rate of health and education expenditure. This is modified only slightly towards the end of the projection period when the extra savings, stimulated by the higher interest rate, slightly reduces the effective tax rate and thus stimulates labour supply. But this is not sufficient to counteract the effect of a higher interest rate on debt servicing costs. The question arises of whether other market responses could modify the debt increase; as mentioned above, these might include general equilibrium effects on wage rates, the exchange rate and relative prices. 6 Furthermore, the various policy instruments modelled here, such as expenditure growth rates and tax rates, cannot provide an endogenous stimulus to the economy, with the exception of a small boost to productivity generated by an increase in the growth of health and education expenditure (which is insufficient for it to be self-financing). As explained earlier, the aim here is to take a small step to endogenise a limited number of responses to policy changes designed to achieve fiscal sustainability. These are examined in the following section. 6. Policies to achieve fiscal sustainability As indicated above, there is a potentially wide array of indicators of fiscal sustainability. The European Commission (2006) has developed and applied a number of indicators, including S1 and S2, defined as follows: S1 measures the constant annual improvement (measured as a proportion of GDP in each period) needed in the fiscal balance in order to achieve a given debt target within a specified time period. This represents 'medium-term' challenges. S2 measures the constant annual improvement (measured as a proportion of GDP in each period) needed in the fiscal balance in order to satisfy the inter-temporal budget constraint over an infinite horizon. Where projections (assuming no policy changes) are made over a finite 'medium term', the debt ratio in subsequent years is assumed to remain constant at its value in the final projection year.
The derivation of these indicators is set out in Appendix 2, where Equation ( B14) corresponds to S1 and Equation (B9) to S2. A property of both sustainability measures is that they ignore the question of whether debt is increasing or decreasing at the end of the projection period.
Estimates of both indicators were made for New Zealand, using the benchmark values of the previous section. In the case of S1, the annual improvement needed in the fiscal balance each year was computed over a 40-year horizon in order to reach a given terminal debt ratio. For terminal debt ratios of 20%, 45% and 60%, the required annual improvements in the fiscal balance (as a percentage of GDP) are found to be 3.6%, 3.3% and 3.1%, respectively. Hence, only a modest additional adjustment to the fiscal balance is needed to achieve a terminal debt of 20 compared to one of 60%. In the case of the infinite horizon (S2), the annual improvement in the fiscal balance would need to be 6.2%.
While these indicators are useful in providing a quantitative measure of the extent to which fiscal policy would need to be adjusted, they have a number of limitations. First, they are not realistic, in the sense that a constant increase in the fiscal balance is not a feasible approach to fiscal management. Governments typically vary tax and expenditure policies in accord with social needs and constraints imposed by prevailing economic conditions. Second, the measures make no reference to actual policy instruments. Third, it is important to know the impact of different policy choices on the time paths of key macroeconomic variables. The following sections, therefore, report the results of a series of simulations for a range of policies. In each case, there is no attempt to specify a precise time path of the debt ratio. Rather, a terminal debt target of 20% is imposed, and the resulting path observed. As the model does not lend itself to finding an analytical solution, the critical values for a particular policy are found by iterating until the 20% debt target is reached.
In examining alternative policies here, no attempt is made to produce any concept of an optimal policy response. This would require the specification of a social welfare, or evaluation, function expressed in terms of a range of performance measures.

Productivity
An improvement in the underlying growth rate of labour productivity would obviously lead to higher rates of economic growth, increased tax revenues and potentially an improved long-term fiscal outlook. It is, therefore, of interest to examine by how much the annual rate of productivity growth would need to increase in order to meet a debt target of 20% in 2053, that is after 40 years? 7 The effect of a higher growth rate is shown in Figure 4 by the time path labelled 'higher productivity'. To achieve this the growth rate, r B would need to rise immediately from its base level of 1.5% to 1.85% annually and remain sustained at this rate over the projection horizon. The debt ratio would remain below its initial level throughout but, as the debt ratio rises toward the end of the period, a higher rate may be need for longer term sustainability beyond the projection horizon. In the absence of feedbacks, the required rate would be marginally higher at 1.88% . 8 It remains a moot point as to whether these productivity increases are feasible, as they lie outside the range of historical experience.
It is unrealistic to expect an immediate increase in productivity growth that could be sustained indefinitely. There are many policies that affect this rate and it would take time for any changes to flow through to higher rates. An alternative case was, therefore, analysed in which the growth rate would, starting from the benchmark value of 1.5%, increase at a slow but constant rate of 0.000297 each year. This would achieve the terminal debt target of 20%, as shown in Figure 4. However, after an initial decline in the debt ratio, it would rise above its starting value before falling to meet the terminal target. Furthermore, instead of a rate of improvement in productivity of 1.88% annually (as in case of a constant level discussed above), the terminal rate would now need to reach 2.65%, which is well outside the range of past experience.
Policies which lead to investments in human capital, through health and education spending, provide a further channel through which labour productivity can be enhanced, as modelled in Equation (21). The question, therefore, arises as to whether there could be a long-term social dividend in the form of higher productivity resulting from additional investments in health and education. To explore this possibility, it was assumed that in the first instance per capita expenditure growth would continue at its historical rate of 2% annually. This would raise the rate of growth of labour productivity from its base rate of 1.5%À1.53%, corresponding to a 2% increase. Were the investment to increase from 2% to 3% the net effect would be to raise labour productivity to just 1.533%.
It is apparent that even with unrealistically high rates of growth of spending on health and education, the impact on labour productivity growth would be minimal. At the same time, the debt ratio would rise as a result of greater public expenditure. This result should not be interpreted as denying the possibility of a return to social investment. Effective investments targeted at specific population groups at risk may well improve their lifetime outcomes and individual productivity in a way that would generate a positive social rate of return. But in using a highly aggregated model, it has not been possible to generate such results. Furthermore, much of this spending is actually annual maintenance (for example, educating each new cohort of school entrants) and public spending is only a part of the total investment that individuals make in their own health and education.

Expenditure policies
Reduced public expenditure is one approach to fiscal sustainability. To attain a terminal debt target of 20%, the per capita growth rates of all categories of government spending in this model would need to be reduced equi-proportionately by 21%. This would imply the growth rates of health and education spending be reduced from their historical level of 2.1%À1.6%, and NZS rates from 1.3% to 1.0%. The path of the debt ratio towards its target level is shown in Figure 4. However, the absolute real levels of these expenditures would still continue to increase over time. Figure 5 illustrates the long-term paths of total expenditure tracks with and without the reduction in per capita growth rates.
One suggested response to population ageing in New Zealand is to increase the age of eligibility for NZS. The growth rate of total expenditure on superannuation is equal to the sum of the growth rate of the payment per eligible person and the growth rate of the eligible population group. Such a policy change, therefore, operates via the latter growth rate. Total expenditure growth on superannuation would, therefore, be expected, depending on the precise response of labour force participation, to fall initially and then increase towards its former level, though total NZS expenditure in absolute terms would remain lower than otherwise.

Taxation policies
This section reports on the implications of a range of options for changes to taxation. They are summarised in Table 2 and the debt tracks are illustrated in Figure 6. In each case the policy is analysed holding all other tax and expenditure policies at their benchmark levels. For example, in the tax smoothing case, the value of t needs to be increased from the benchmark of 16.25% to 18.5% in each year, when allowance is made for feedback effects, which are here dominated by the adverse incentive effects of taxation. 9 Not allowing for the feedbacks would suggest a lower increase to 18.0% each year. Delayed tax smoothing produces less variation in the debt ratio over the projection period. Indeed, with an immediate increase in the tax rate, there are surpluses over a period of around 20 years. Furthermore, at the end of the projection period, the debt ratio continues to increase relatively sharply, suggesting that additional adjustments to the tax rate will be needed.
The fact that tax smoothing produces a period during which there is a surplus gives rise in practice to the temptation to spend part of the surplus. That is, the tax policy produces a possible endogenous expenditure change which governments often find difficult to resist. 10 This is of course just one consideration in evaluating alternative policies and, in particular, intergenerational comparisons are relevant. However, these aspects cannot be considered here.
If, instead of smoothing, the percentage tax rate were to be increased by 0.14 each period (so that it becomes 21.9% in 2053), the target debt ratio can be achieved. In the case of delayed tax smoothing and delayed annual increase, the benchmark income tax rate is held constant for the first 10 years of the projection period. The variation in the debt ratio over the projection period is the lowest in the case where the tax rate is gradually increased from the beginning of the period.
When a gradual tax increase is delayed until a debt threshold of 35% of GDP is reached, the first change in the tax will take place in the year 2034. Not surprisingly, the annual increase and the final tax rate needed to achieve the 20% debt target in 2053 is now much higher than when action is taken  earlier. In addition, it implies higher intermediate debt ratios. The fact that tax rates are ultimately higher also means that the adverse incentive effects are greater. This means that the difference between the required tax adjustment with no-feedbacks and those allowing for feedback effects is also much higher: the rates differ by four percentage points in 2053. Consider further the profile of the debt ratio in the case where the gradual increase in the income tax rate is delayed until 2034. The projections show that a steady increase in the tax rate can achieve a 20% debt ratio by 2053, the end of the period. However, the debt ratio continues to increase until around 2043 so that, without longer term projections, it may be thought during this period that the tax rate should be increased even faster À it would not be evident that the profile will turn down towards the end of the period.
The profiles in Figure 6 allow for the various feedback effects, the most important of which concerns adverse incentive effects of taxation. The tax policies all ensure that the debt ratio, despite variation over the period, remains within a reasonable range of the target value. This means that the risk premium remains relatively steady. In the absence of feedback effects, the main implication is that both the income tax rates and, where relevant, their annual increases are lower, as indicated in Table 2. However, the fact that the resulting debt ratio profiles intersect at the start and end dates means that the ratios for intermediate years, whether or not allowing for feedbacks, do not deviate significantly from each other.
It is important to recognise that this finding should not lead to the conclusion that feedbacks have a minor influence. If the economy is allowed to accumulate very high debt ratios, then considering tax and expenditure policy changes that do not allow for feedbacks will give much too optimistic a view of what is needed. If, from a high debt position, a policy change does not prevent the economy from moving into the range where the risk premium rises sharply, severe problems can arise from high debt servicing costs. This interest rate problem is exacerbated by large changes in taxation, which give rise to strong adverse incentive effects. It is very hard to reverse severe problems À perhaps leading to default. But the no-feedback case allows the economy to move through periods of very high debt ratios and reduce debt with sufficiently large tax increases. It appears, incorrectly if the feedbacks are ignored, that a large degree of intergenerational redistribution is able to get the economy out of trouble.

Conclusions
Ageing populations are leading to long-term pressures on government budgets in many countries; New Zealand is no exception. In the medium term, the challenge has become even more marked as countries recover from the global financial crisis and endeavour to restore their fiscal balances and reduce public debt levels. This paper develops long-term projections as the starting point for analysing options to achieve long-term fiscal sustainability. Importantly, it was seen that the lack of longterm sustainability arises primarily from demographic changes rather than fundamental problems with tax and expenditure design settings. A principal focus has been on incorporating some selected economic feedbacks into a demographically driven model of government debt.
For example, rising debt levels could be expected to influence interest rates paid to foreign holders of New Zealand dollar denominated securities. Furthermore, higher tax rates could have disincentive effects on labour supply. When fiscal policy responds to ensure long-term sustainability, these feedbacks can potentially modify the intended outcomes by enhancing or dampening the effect of the policy interventions.
A small model is developed that captures, at a high level of aggregation, the evolution of public expenditure, tax revenue and public debt levels. It is used to project the key outputs over a 40-year period. In the first instance, the model, excluding any feedback effects, is calibrated and shown to track the long-term debt path of the highly detailed LTFM used by the Treasury. Following the incorporation of a small number of feedbacks, the model is used to test the effect of policy changes with a view to achieving a net debt target of 20% of GDP after 40 years. It is shown, for example, that in the case of tax changes, the presence of the feedback effects implies that a greater increase in income tax rates would be needed to achieve the same debt outcome than in their absence.
The achievement of a specified debt ratio target by the end of the projection period is rather arbitrary and is used purely to illustrate the different debt paths taken as a result of different policies. In particular, the different policies were seen to imply very different debt profiles at the final projection year, with some (such as tax smoothing) implying a large rate of increase while others (such as a delayed tax increase after a threshold debt ratio is reached) implying a rapid decrease in year 40, and others (such as a gradual tax increase) involving a much smaller rate of change.
A potentially significant avenue for achieving fiscal sustainability is raising the average annual rate of labour productivity growth. However, there is no suggestion that this would be easy and it may involve other spending decisions. The model is used to demonstrate the critical importance of population ageing for fiscal sustainability. If the total size of the population were to grow at the projected rate but the age composition were to remain unchanged, fiscal sustainability would be assured without further policy responses. Reducing the rate of growth in public expenditures could lead to a sustainable fiscal position even though total absolute expenditure would continue to grow in real terms.
The present model can be extended in two important ways. First, it has been assumed À in common with the majority of projection models À that the relevant parameter values, growth rates and so on, are known with certainty. However, uncertainty can be introduced into the model, allowing for the production of probability distributions of the debt ratio in each year. 11 This extension has been made by Ball, Creedy, and Scobie (2016) who show, among other things, that the deterministic projections provide a reasonable guide to the median values obtained from stochastic projections.
Second, in examining alternative policies to achieve a desired debt ratio in the final projection period, no consideration has been given to any concept of an optimal policy response. It would be necessary to introduce a multi-period social welfare, or evaluation, function, rather than simply aiming for a given debt target in one year. Nevertheless, the model provides a strong foundation for further extensions. In particular, incorporating uncertainty can provide a richer context for the analysis of optimal policy. This is because the combination of uncertainty with the fact that costs of immediate action (in the case of, say, tax smoothing) are non-reversible can lead to the existence of an 'option value' of waiting until some of the uncertainty is resolved. These extensions are planned for future research. Notes 1. A similar approach was adopted by the Australian Treasury (2015). Variants of this kind of procedure were also used to examine projected New Zealand social expenditures only, although allowing for stochastic elements, by Creedy and Scobie (2005) and Creedy and Makale (2014). 2. Kleen and Pettersson (2012) include labour supply effects using an elasticity of the employment ratio with respect to the tax rate. They also assume that productivity falls slightly as labour force participation increases (on the argument that the new entrants to the labour force resulting from a tax cut are relatively less productive). 3. In the US context, Jorgenson and Stiroh (2000) found improvements in the quality of labour accounted for nearly 15% of labor productivity growth for the period 1959À1998. 4. It may, in addition, be thought that productivity change may be influenced by changes in the interest rate. However, this effect is likely to come via possible higher investment resulting from reductions in the interest rate. The elasticity of r with respect to r can be expressed as the product of the elasticity of r with respect to investment, and the elasticity of investment with respect to the interest rate. The overall effect is likely to be very small, and is therefore ignored here. 5. In view of the generally low debt ratio in New Zealand, it may be argued that the risk premium would begin to increase more rapidly before ratios as high as 150% are reached. Some sensitivity analyses relating to risk-premium parameters are reported in Appendix 3. 6. For an extensive discussion, which cautions against an excessive concern for population ageing, see Disney (1996). 7. Wilkinson and Acharya (2014), using the Treasury's Long-term Fiscal Model (2013c), estimated that if the base rate of annual productivity growth of 1.5% could be raised to 1.94%, a debt target of 20% could be reached by 2022 and maintained at that level, without any reduction in real per capita aggregate spending. However, their experiment did not use the 'benchmark', or expanding debt, projection but the 'Sustainable Debt' scenario of the LTFM. 8. Treasury (2013a, p. 16) takes a less benign view about the effects of an increase in productivity, on the argument that there would be pressures for higher spending, arising, for example, from the link between NZS and wage growth. 9. It may be suggested that in the tax smoothing case the government could use the initial fund to obtain a higher rate of return than the borrowing rate. This would potentially alleviate some of the long-term fiscal pressure. However, in this type of model and growth models generally, it is usual to assume that lending and borrowing rates are the same. To do otherwise adds considerable complexity. 10. Davis and Fabling (2002) model 'expenditure creep' and report that it can completely erode the efficiency gains from tax smoothing. They conclude that 'strong fiscal institutions are a prerequisite for achieving the welfare gains from tax smoothing' (2002, p. 16). 11. Such distributions are essentially conditional distributions. By allowing various growth rates to be stochastic, projections are conditional on the model specification itself. 12. In European Commission (2006), this is decomposed further as above using DB t D B t ¡ B 0 .

Appendix 1. Further details of model calibration
This appendix provides details of the benchmark calibration values and data sources. These are summarised in Tables A1ÀA6. Data for Table A3 Using P T ¡ 1 j D 0 (1 C r 0 ) j D {(1 C r 0 ) T ¡ 1}/r 0 , the required B Ã T is given by 12 The value of B Ã T indicates the extent to which government expenditure or tax revenue, or a combination of both, must be adjusted each year in order to attain the debt target ratio, D 0Ã T , in the final projection year.

Appendix 3. Some sensitivity analyses
There is insufficient space here to report all sensitivity analyses, but Table A7 provides information about the implications of varying some of the basic parameter values used in the feedback functions. The table provides the terminal value of the debt ratio for high and low values of the relevant parameter, as well as the benchmark value. The full set of parameters and corresponding formulae are listed in Table 1.
The parameter, u 1 , of the risk premium relationship has the effect of shifting the whole schedule up or down. Hence its variation involves systematic changes in the interest rate over the whole period. The risk premium clearly affects both interest-income tax revenues as well as debt repayment costs. The productivity growth rate parameters, u 5 and u 6 , affect the curvature of the sigmoid logistic relationship between the change in the growth rate of expenditure on health and education and the change in the productivity rate.
In generating these sensitivity results, substantial changes in the parameters were chosen deliberately in order to illustrate the robustness of the model. Despite these changes, the impact on the terminal debt ratio is typically modest. It should, nevertheless, be borne in mind that variations in only one parameter are expected to have relatively small effects on the ultimate debt ratio because the basic simulation involves constant growth rates and constant tax rates. Some indication of the effect of simultaneously changing several parameters can be obtained as follows. Suppose all the parameters in Table A7 were set at the values giving the lowest final debt ratio. This produces a final ratio of debt to GDP of 122. Alternatively, if all parameters are set at levels associated with higher debt ratios, the final ratio increases to 189.