2  Analytical framework

2.1  The intertemporal budget constraint (IBC)

The framework used here draws on the specification of the IBC used by Buiter (1995, 2001), Wells (1996) and Romer (2001). The specification is stylised in that it excludes capital spending and income from government trading enterprises. The government budget identity is given by:

(1a)    

where G is current spending on goods, services and transfers, T denotes taxes, and r is the real interest rate. For simplicity, r is assumed to be constant and there is no inflation. B is the end-of-period stock of debt and t denotes a time period (where t would denote a year if G and T represented annual flows). In equation (1a), government current spending, including debt servicing is financed from taxes and changes in debt. Rearrangement gives the equation for the evolution of debt between period t and t-1 (i.e., over one year if t denotes a year):

(1b)    

The government’s primary balance, PB, excludes all transactions involving debt and is defined as:

(2)    

A positive value of PB_t indicates a primary surplus arising from the flow of T and G transactions during the period t. The overall fiscal or “operating” balance for period t(OB_t) is defined as:

(3)    

The primary balance isolates the underlying spending and tax paths from any dynamics created by the interaction of the interest rate and debt (or financial assets). This is important in the context of the fiscal gap because it is assumed that the fiscal authority can directly influence T and/or G and change them if necessary to satisfy the IBC.

The IBC is concerned with the future paths of fiscal and economic variables. By repeated substitution, debt at time N (i.e., B_N) is a function of its initial value and the entire series of primary balances realised between the initial period and N. If the stock of outstanding debt grows at a rate less than r, then in the limit, the present discounted value of terminal debt goes to zero as time tends to infinity:

(4)    

If this terminal condition is satisfied, then the government’s IBC holds if the excess of primary surpluses over primary deficits, in present value terms (the right-hand-side of equation 5), matches the value of outstanding debt (B₀):

(5)    

This expression is sometimes described as the government’s present value budget constraint (PVBC). Now let debt and the primary balance be defined as ratios-to-output, b and pb respectively, where b_t = B_t / Y_t , pb_t = PB_t / Y_t and where Y_t is real output in period t. Assume that the economy operates with a long-run real interest rate above the long-run real growth rate of output (g), where the latter is also assumed to be constant.[4] If:

(6)    

then an analogous version of equation (5) is given by:

(7)    

2.2  The fiscal gap

Following Auerbach (1994), the fiscal gap can be defined as the immediate and permanent change in the primary balance (brought about by tax changes and/or spending changes) that, if projections prove accurate, would be needed to bring the debt-to-gross domestic product (GDP) ratio at some date T in the future to the level that prevails at some initial date t (see Section 3 below for further discussion).

The fiscal gap can be “closed” through changes in taxes (T) and/or (non-interest or primary) spending (G) that permanently alter the primary balance. Closing a fiscal gap does not imply eliminating any current difference between taxes and primary spending, as the primary balance may be non-zero in the initial year.

Based on Auerbach (1994, p.170), and using the notation from above, the change in the primary balance as a share of GDP (Δpb) required to eliminate the fiscal gap is equal to:

(8)    

where T denotes the terminal year, t is the initial year and pb_s is the primary balance as a share of GDP in year s. A positive value for Δpb from equation (8) implies that a permanent increase in the primary surplus (or a permanent reduction in the primary deficit) is needed to ensure that the initial debt-to-GDP ratio (b_t) is attained in the terminal year. A negative fiscal gap means that a permanent reduction in the primary balance is consistent with returning to the initial debt-to-GDP ratio in the terminal year. A fiscal gap of zero indicates that the projected path of the primary balance is such that the initial debt-to-GDP ratio is attained in the terminal year.

If the real interest rate exceeds the real growth rate of output (i.e., if (r – g) > 0), and there is some initial debt, then pb_s = 0 for all periods beyond t will imply a non-zero fiscal gap. In these circumstances, expression (8) reduces to Δpb = (r – g)b_t. A primary balance of zero will see debt servicing add to debt. With r > g debt servicing will add to debt faster than the rate of output growth and the debt ratio will increase. Primary surpluses will be required at some point to stabilise the debt ratio.

The fiscal gap indicator is similar to the “tax gap” set out in Blanchard (1993). The tax gap is the difference between the actual tax rate and a constant tax rate that ensures fiscal sustainability in terms of the IBC.[5] The fiscal gap is also similar to the measure of “Economic Net Worth” (ENW) set out in Wells (1996). ENW is defined by the present value difference between a time-H forecast debt ratio and a desired debt ratio. ENW is zero if on unchanged fiscal policy the path of primary surpluses is consistent with the desired debt ratio at the terminal date. The methodology allows calculation of the proportional change to the sequence of primary balances needed to meet the terminal date debt ratio.

Recent work by the OECD on the fiscal implications of population ageing has considered a similar measure. For a “stylised” OECD country (i.e., one which has the features of the median OECD country) the analysis calculates the change in the primary balance needed to ensure a broadly unchanged debt-to-GDP ratio at the end of the projection period (2050) (see OECD, 2001; Dang, Antolin and Oxley, 2001).

Calculating a fiscal gap using equation (8) requires projections of future primary balances and projections of future interest rates and growth rates. These elements are discussed in Sections 3 and 4 with reference to overseas applications of the methodology.