As noted in this question here, the shutdown has resulted in a 17% reduction in spending:
Based on estimates drawn from CBO and OMB data, 83 percent of government operations will continue. This figure assumes that the government pays amounts due on appropriations obligated before the shutdown ($512 billion), spends $225 billion on exempted military and civilian personnel, pays entitlement benefits for those found eligible before the shutdown (about $2 trillion), and pays interest costs when due ($237 billion). This is about 83 percent of projected 2014 spending of $3.6 trillion.
This means we are spending $2,988 billion this fiscal year (Oct2013-Sept2014) instead of $3,600 billion, or a savings of $612 billion in FY2014. According to CBO figures in July, we will decrease the deficit from $642 billion to $30 billion. During the shutdown, we will increase our debt at a rate of $30 billion per year instead of $642 billion (or as a percentage, 0.18% instead of 3.84%)
If the current laws that govern federal taxes and spending do not change, the budget deficit will shrink this year to $642 billion
PolitiFact says that the 85% estimate is plausible.
In order to reach Paul’s 85 percent threshold, the government would need to be spending $404 billion from its "discretionary" (or appropriated) budget beyond military personnel. That means the government would be spending 43 percent of the remaining discretionary spending on activities that continue through the shutdown.
Experts we spoke to said that percentage is plausible. Even if you cut that $404 billion figure in half, Paul would still be pretty close -- the figure would be 79 percent rather than 85 percent. We’ll only know the final number for certain after the shutdown is over.
If you assume that the economy will be affected by the shutdown the same way it was during the 1995-1996 shutdown, you have to reduce the savings estimate by $27 billion. This would mean that the deficit would be reduced by $585 billion, and the debt would increase $57 billion instead of the projected $642.
To figure a day to day savings, multiply these figures by (1/365). To figure total savings during the entire shutdown (multiply by #days shutdown/365). If employees get paid retroactive back pay for not working, ending the shutdown will cost $27b * (#days shutdown/365).