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Appendix 4. Calculation of the Sustainable Non-Oil Deficit

Niko Hobdari, Eric Le Borgne, Chonira Aturupane, Koba Gvenetadze, John Wakeman-Linn, and Stephan Danninger
Published Date:
April 2004
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The paper discusses two long-term strategies for the use of oil assets. The first is based on a fixed expenditure rule, which—together with a permanent financing condition constraint—pins down the time path for the sustainable non-oil deficit. It assumes that the government will ensure that a predetermined expenditure path can be permanently financed out of an accumulated financial asset stock. The second strategy aims to preserve oil wealth indefinitely and thus creates a permanent income and expenditure stream. This strategy requires adherence to a long-term savings path. The level of available resources for consumption, that is, the sustainable non-oil deficit path, derives as a residual.

The presented scenarios examine the long-term implications of these two strategies. The calculations explore the direct effect of changes in crucial parameters such as the oil price of the real interest rate, but do not model their interactions with other variables such as the growth or inflation rate. The following two sections discuss the computation of the sustainable deficit under the two strategies.

A. Constant Real Expenditure or Constant Real Per Capita Expenditures

The government guarantees financing of constant expenditures in real terms or in real per capita terms (alternative scenario):

Et = Et–1(1 + p) = E0(1 + p)t,

where p denotes either the long-term inflation rate or the population cum inflation growth rate. E0 denotes the initial level of expenditures. The sustainable non-oil deficit is defined as the corresponding financing need for this consumption path. In order to secure this financing stream a sufficiently large stock of financial assets FW needs to be saved by period T when the revenue stream ceases (RT = 0), so that expenditure financing can continue. The permanently fundable level of real expenditure from FWT is the real interest earnings on the assets stock:

Et = FWt i for all tT.

The formation of the financial wealth FWt follows the dynamic process

FWt = FWt–1 (1 + i + p) + (RtE0 pt) α (1 + i + p).

where the fraction a denotes interest earned on within-year flows and was set to α = ½, assuming a constant rate of within-period net flows. At a given revenue stream and interest rate, the level of financial assets is determined by the initial level of expenditures. Thus the maximum E0 that can be chosen has to be sufficiently low to generate a financial wealth stock by period T, which allows permanent financing of the expenditure path, or

E0 (1 + p)T = FTr = ET.

In order to solve for the maximum sustainable deficit path, the following iterative process can therefore be applied:

  • Choose a level for E0;
  • Determine FWT (E0);
  • Determine ET(1 + p);
  • Check whether the permanent financing condition is met.

If E0 (1 +p)T < FT (E0) rT, increase E0 and go back to step 2.

If E0 (1 +p)T > FT (E0)rT, decrease E0 and go back to step 2.

B. Constant Real Wealth and Constant Per Capita Wealth17

This strategy aims to preserve oil wealth in order to generate a permanent income flow. In particular it requires that total oil wealth TW (or per capita wealth) remains constant for all periods and thus enables a permanent income stream. Total oil wealth TW is defined as the value of physical oil assets OWt in the ground plus the value of financial assets FWt created from savings out of oil revenue:

TW = OWt + FWt.

The stock of oil wealth in the ground is given as the present discounted value of future revenue streams and declines over time as reserves in the ground are depleted:

Financial wealth in period t is determined by interest earnings and additions from new savings Si.

FWt = (FWt–1 + St)(1 + i).

As oil wealth gradually declines over time, financial wealth and thus savings have to adjust to ensure that total wealth (per capita wealth) remains constant. Therefore the condition TWt = TWt–1 implies that

OWt–1OWt = FWtFWt–1.

From above it is known that

OWtOWt–1 = i OWt–1Rt


FWtFWt–1 = i FWt–1 + St (1 + i),

so that it is now possible to solve for the required savings path Si.

St = 1/(1 + i){Rti (OWt–1 + FWt–1)}.

The path of permissible expenditures is then given by Et = RtSt,, which represents the sustainable expenditure ceiling.


For a detailed derivation see, for instance, Davoodi (2002).

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