Given calculations of government revenue and expenditure, it is possible to compute the government balance, adjusted by foreign aid donations when given (for donors, the sign of AID is negative). Repayments of loans to the IMF or World Bank also reduce the balance.
That allows the update of absolute government debt and a calculation of its magnitude relative to GDP. We use the relative price adjusted GDP because it reflects changes in energy and agricultural prices that can be of considerable significance for countries with heavy dependence on trade in those sectors.
It is government debt as a percentage of GDP that IFs uses to build equilibrating dynamics for government revenues and expenditures.
In years beyond the first, the government debt as a percent of GDP will give rise to pressure for higher or lower levels of government revenues and expenditures. Those pressures will be conveyed via two multipliers applied to calculations of taxing and spending in subsequent years. Those multipliers are set at "1" in the first year, indicating no change in pressures initially. The rest of this section explains how those multipliers are changed over time.
The first step in feeding back government debt level to pressures for increases or decreases in revenues and expenditures is to compute a target for government debt as a percent of GDP with which to compare the actual level. A function relating typical levels of debt to GDP per capita is used as the foundation of that target. Dynamically, the target begins as the initial ratio of debt to potential, relative price adjusted GDP, but we found that it enhanced long-term model behavior to set the target as a moving average of that ratio over time, only allowing the moving average to be changed over time when the debt to GDP ratio falls (thus reducing the target). Further the target converges to the minimum of the moving average and the value from the function.
The tricky part is to get a dynamic system to chase the target over time, adjusting revenues and expenditures annually as it does so. IFs does that in this instance and in others by using an adjustment to feedback parameters based on two terms: an integral term (the absolute distance of the system from the target) and a differential term (the change in values of the target relative to the preceding year). Engineers refer to this process as a PID controller. The two terms are computed as Diff1 and Diff2. A potential relative price adjusted GDP term is used rather than the actual GDP of the year in order to protect systemic stability over time (use of GDP in the target term can set up oscillations in some feedback loops also involving GDP).
Once the two terms are available, the PID adjuster routine (ADJUSTR), described elsewhere is called by IFs to convert the difference terms, modified by exogenous parameters, into a multiplier on the cumulatively-computed revenue multiplier term used in equations above. (This function historically used the parameters elgrevdebt1 and elgrevdebt2 , but those are currently hard-coded in the equation below as 0.15 and 0.30).
In the above specification, the adjuster uses two elasticities for the difference terms. In completely parallel fashion, adjustment is made to the expenditure multiplier. Tuning of the model reinforced the need for the elasticities on the expenditure side normally to be lower than those on the revenue side (governments are more likely to adjust revenues than expenditures). Elgexdebt1 and elgexpdebt2 were set at 0.05 and -0.1, respectively. Yet when revenues and expenditures get substantially out of balance, there are times that some "extra kick" on the expenditure adjustment is needed and that is provided by the elasticity multiplier (ElMul) based on the ratio of revenues an expenditures.