Let the change in spending be
, the tax rate be
, and the multiplier rate be
, then the change in revenue,
, is given by the equation:
As a result of this spending, the overall cost or savings,
, can be obtained from the changes in spending and revenue according to the equation:
Furthermore, let the real interest rate be
. Then the resultant change in interest payments,
, can also be determined from the costs and the interest rate:
Finally, we can use this last calculation and the tax rate
to calculate the change in GDP, ΔGDP
, required to either nullify the proposed fiscal savings in the case of a spending reduction or nullify the possible fiscal costs in the case of a spending increase:
These four equations are used to calculate the revenue increase/reduction,
, the cost or savings,
, the change in interest payments,
, and the change in GDP,
, sufficient to eradicate the fiscal effects of the original cut or increase in government spending,
. All results are given as a percentage of the GDP. These five macroeconomic effects are then displayed as a bar chart with negative values labeled to the right in red and positive changes labeled in blue.
The model is initialized with the values discussed by Krugman. He makes the case that any austerity measures, including a reduction of government spending, will result in only a negligible reduction of the real interest payments that affect the short-term deficits. Furthermore, such a reduction in spending will lead to a loss of future revenue and a possible reduction of the GDP. If the GDP drops by even an insignificant amount of .08%, then the fiscal savings as a result of the austerity measures will have been erased, thereby undermining the reason for the whole enterprise. This argument can also be run in reverse. If government spending is increased, then this causes a corresponding increase in revenue, which will likely lead to an increase in GDP. The cost for such a measure is accompanied by only a very small increase in interest payments, which add negligibly to the short-term deficit. In addition the increase in GDP required to offset the initial costs is only a small percentage of the GDP, usually less than 0.1 of one percent.
The implications of this simple first-order macroeconomic argument, which assumes by the way that the tax, marginal interest, and multiplier rates all remain invariant over the period of analysis, are remarkably robust to enormous variations in these parameters, as can be discovered by playing with the control settings. The evidence for the model rests in the Depression-era history of the United States and Europe in which Keynesian pump-priming led to significant growth in the GDP of each country, which easily offset the financial costs. We are in a similar situation today, so that the mathematics of this model destroys the usual austerity argument and strongly supports the notion of further stimulus as a means to economic recovery.
P. Krugman, Self-Defeating Austerity, The Conscience of a Liberal, New York Times
, July 7, 2010.