Debt sustainability modeling

As emphasized by Chalk & Hemming (2000), most of the analytical discussions on debt sustainability revolve around a representative agent model in which the government satisfies two types of constraints in order to preserve fiscal solvency: static budget constraint in each period and intertemporal budget constraint in the long-term horizon. While intertemporal budget constraint approach provides an important theoretical background of the fiscal sustainability concept, its empirical application is possible only to

77 certain extent. In this section I provide theoretical framework of the fiscal sustainability concept which utilizes intertemporal budget constraint.

Within the fiscal sustainability theoretical framework, intertemporal budget constraint can be derived from the static budget constraint. Static budget constraint in the context of the debt accumulation equation corresponds to the DAE formulation given in (2.8), that value of current debt equals sum primary balance, current interest payment and debt from the previous period. A derivation of the intertemporal budget constraint presented in this work follows debt accumulation in relative terms, based on Giammarioli et al. (2007), but it can be derived also in absolute terms. If equation (2.8) is rewritten in relative terms as

𝐷𝑡 𝑌𝑡 = −𝑃𝐵𝑡 𝑌𝑡 + (1+𝑖𝑡)𝐷𝑡−1 (1+𝑔𝑡)𝑌𝑡−1 (3.1)

public debt dynamics can be considered as a sum of current fiscal stance −𝑃𝐵𝑡

𝑌𝑡 and

inheritance of past fiscal policies (1+𝑖𝑡)𝐷𝑡−1

(1+𝑔𝑡)𝑌𝑡−1. From the equation (3.1) follows that

stabilization of the 𝐷𝑡

𝑌𝑡 over time in case when interest rate 𝑖𝑡 is higher than GDP growth rate

is possible only if primary balance is proportionally in surplus. However, static budget constraint is an accounting identity and does not impose any restriction on fiscal stance until lenders are willing to finance primary deficits. Nevertheless, if the government keeps running fiscal deficit for the prolonged period of time, additional borrowing will only be possible if lenders are confident that government finances will remain sound and solvent in the long run. Therefore, intertemporal budget constraint provides the answer on the question what restrictions on current and future fiscal policies should be satisfied to maintain long-term debt sustainability. The starting point in derivation is rearrangement of (3.1) by moving lagged debt to the left-hand side of the equation:

𝐷𝑡−1 𝑌𝑡−1 = (1+𝑔𝑡) (1+𝑖𝑡)( 𝑃𝐵𝑡 𝑌𝑡 + 𝐷𝑡 𝑌𝑡). (3.2)

From the (3.2) follows that at any point in time value of debt-to-GDP can be expressed recursively as a function of one-period ahead primary balance and debt, 𝐷𝑡

𝑌𝑡 = (1+𝑔𝑡+1) (1+𝑖𝑡+1)( 𝑃𝐵𝑡+1 𝑌𝑡+1 + 𝐷𝑡+1

𝑌𝑡+1). Lets assume that debt sustainability is examined over certain future

period of time [1, … , 𝑇 − 1]. In such case, initial value of the debt-to-GDP can be recursively written as 𝐷0 𝑌0 = (1+𝑔1) (1+𝑖1) 𝑃𝐵1 𝑌1 + (1+𝑔1)… (1+𝑖1)… (1+𝑔𝑇)𝑃𝐵𝑇 (1+𝑖𝑇)𝑌𝑇 + (1+𝑔1)… (1+𝑖1)… (1+𝑔𝑇)𝐷𝑇 (1+𝑖𝑇)𝑌𝑇, (3.3)

78 where 𝐷𝑇 referes to the terminal value of debt. The intuition behind this equation is that

current value of debt in equilibrium long run case equals sum of discounted primary balances in each period and sum of discounted terminal value of debt, where (1+𝑔1)…

(1+𝑖1)… (1+𝑔𝑇)

(1+𝑖𝑇)

is respective discount factor for the period t. If for a sake of simplicity discount factor is denoted as 𝜌_𝑡 and written in recursive manner as

𝜌_𝑡 =(1+𝑔1)

(1+𝑖1)𝜌𝑡−1, 𝜌0 = 1 (3.4)

then the equation (3.3) can be simplified to expression

𝐷0 𝑌0− 𝜌𝑇 𝐷𝑇 𝑌𝑇 = ∑ 𝜌𝑡 𝑇 𝑡=1 𝑃𝐵𝑡 𝑌𝑡 . (3.5)

The left-hand side of this equation can be interpreted as the net present value of the debt. The net present value of the public debt will be either positive if the sum of discounted primary surpluses exceeds a sum of discounted primary deficits or negative in opposite case. Yet, the equation (3.5) still does not impose debt sustainability condition since the lenders can be confident about government ability to service debt beyond the time period [1, … , 𝑇 − 1]. Therefore, in the next step limited long-run horizon of the analysis is extended up to infinity, as given in the equation bellow

𝐷0 𝑌0 − lim𝑇→∞𝜌𝑇 𝐷𝑇 𝑌𝑇 = ∑ 𝜌𝑡 ∞ 𝑡=1 𝑃𝐵𝑡 𝑌𝑡 . (3.6)

In the similar manner, intertemporal budget constraint in absolute terms reads as (Chalk and Hemming, 2000):

𝐷₀ − lim

𝑇→∞R𝑇𝐷𝑇= ∑ R𝑡

∞

𝑡=1 𝑃𝐵𝑡, (3.7)

where R_𝑇 is discount factor R_𝑇= 1

(1+𝑖1)R𝑡−1. Both equations (3.6) and (3.7) tell that the

intertemporal budget constraint can be satisfied even in case that discounted value of terminal debt is positive if government runs primary deficits forever by rolling its debt over and borrowing to finance its deficits. However, running primary deficits forever is basically a Ponzi scheme and such outcome is not feasible in case of the finite number of agents, as shown by O’Connel & Zeldes (1988): if some debtor holds government debt forever, she will have lower consumption in at least one period and consequently lower welfare when compared to a situation wherein she does not hold debt at all. Since rational

79 agents are not willing to keep government debt forever if the government is running a Ponzi scheme, a no-Ponzi restriction lim

𝑇→∞𝜌𝑇 𝐷𝑇

𝑌𝑇 ≤ 0 (or 𝑇→∞lim R𝑇𝐷𝑇≤ 0 in case of absolute

debt) is regarded as a necessary condition to maintain fiscal sustainability. If a no-Ponzi restriction is taken into consideration, fiscal sustainability can be operationalized in relative terms as 𝐷0 𝑌0 ≤ ∑ 𝜌𝑡 ∞ 𝑡=1 𝑃𝐵𝑡 𝑌𝑡 , (3.8) or in absolute terms as 𝐷₀ ≤ ∑∞_𝑡=1R_𝑡𝑃𝐵_𝑡. (3.9)

The fiscal sustainability condition imposes that fiscal policy has to respect present value budget constraint, i.e. that fiscal policy is sustainable if the present value of the primary balances (to GDP) is greater than or equal to the current level of public debt (to GDP). Representation of the fiscal sustainability condition based on intertemporal budget constraint provides an important theoretical background for the debt sustainability assessment, but also has several limitations (Jha, 2012): it is difficult to apply, government does not have sufficient control over the future revenues, and historical data has limited usability for long-term predictions. These limitations further impose difficulties in practical application of conditions in (3.8) and (3.9) for several reasons:

 Since the sustainability conditions hold for an infinite time, one can always argue that any short- to mid-term problems with large deficits can be offset by sufficiently large primary surpluses in the future, and vice versa.

 The sustainability conditions do not impose any constraints on the structure and relationship between public revenues and expenditures.

 If fore some reason lenders are reluctant to buy debt in the short to mid run, government may experience serious illiquidity or insolvency issues even if long- term fiscal sustainability condition is satisfied. Such example was the Mexican sovereign default in 1995, although Mexico’s debt in 1993 was quite low at 30% of GDP (Jha, 2012).

In regard to the difficulties for the practical application of the public debt sustainability condition, numerous empirical-based methodologies with greater focus on the mid-term sustainability (like the abovementioned IMF DSA), have been proposed in the literature. They will be discussed in the literature review section.

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