Proponents and opponents of austerity in the economics blogosphere have been hurling recriminations at each other in recent weeks.

Now one of the theoretical arguments in favour of government spending is what is called the "Keynesian multiplier" which shows that a dollar spent by the government results in total spending which is several times higher.

Assume that the government hires unemployed resources to build a $1,000 woodshed. The carpenters and lumber producers get an extra $1,000 in income. If they all have a marginal propensity to consume (MPC) of 2/3, they will spend $666.67 on new consumption goods. The producers of these goods will now have extra incomes of $666.67. If their MPC is also 2/3 they in turn will spend $444.44. The process will go on with each round of spending being 2/3 of the previous round. Thus a chain of secondary consumption spending is set in motion. But although it is an endless chain, the spending adds up to a finite sum. Mathematically, it is equal to 1/(1-MPC) or 3. Thus the $1,000 results in spending of $3,000.

In the US the saving rate is about 5%, so the multiplier by this argument ought to be 20. In reality some of the money spent by the government will later be spent on imported goods and some will be taken back by the government in taxes. Even then the multiplier should be about 15. However, even the most optimistic calculated values of the multiplier are not usually more than 1.5. Why should this be so? Why does government spending have so little effect on the GDP?

The reason is that there is a fundamental flaw in the Keynesian multiplier argument. It confuses the average propensity to consume with the marginal propensity to consume.

In the figure below which shows the saving rate from 1990 to 2015 it will be seen that during each of the recessions the saving rate goes up and stays elevated for several years thereafter. During a recession, by definition, income is falling. A higher saving rate during a recession means that consumption is falling even faster than income. Since both are negative but the fall in consumption is greater than the fall in income, the marginal propensity to consume is greater than 1. Therefore the Keynesian multiplier which is 1/(1-MPC) is negative.

In physical terms this means that when the government spends an additional $1,000 consumers save more than $1,000. After the recession the multiplier moves closer to zero and then into positive territory. But this probably takes several years by which time the justification for government spending is over.

The above is adapted from my new ebook Macroeconomics Redefined