A rising government deficit will always allow the household sector to increase its saving in nominal terms
Answer: True
The answer is True.
Spending equals income equals output - which is a fundamental fact arising from the way the national accounts measures economic activity.
Nominal GDP will always increase if governments inject more net spending into the economy.
As household saving is a positive function of disposable income, which is a positive function of nominal GDP, it follows that household saving will rise in nominal terms whenever GDP rises.
But that does not mean it rises in real terms. If the rising deficit only induce an inflationary response, nominal GDP rises but real GDP does not. So in real terms, household saving might not increase at all (and even fall).