Will a payroll tax cut stimulate the economy? I am going to answer this in the context of Gauti B. Eggertsson' paper "What Fiscal Policy Is Effective at Zero Interest Rates?" where this question is addressed directly (the analysis begins on page 13). The model is New Keynesian.
The answer, in this model anyway, is that in normal times a payroll tax cut would be stimulative, but at the zero bound it's not so clear. Let me see if I can explain why.
When interest rates are positive, the framework is essentially a standard AS-AD model:
A payroll tax cut increases labor supply and shifts out the AS curve. The shift in the AS curve results in lower inflation and higher output/employment.
One thing that is left out of this model to simplify the analysis and keep it tractable is the demand-side effects of such policies. That is, a tax cut would also increase AD. If we add this effect, the graph then looks like:
Output goes up even more, but whether inflation goes up or down depends upon which shift is larger, the shift in the AS or the shift in the AD (based upon the evidence on how labor supply responds to changes in taxes, I would expect that the shift in the AD would be larger and come first, but ultimately that is an empirical matter).
When the economy is at the zero bound for the nominal interest rates things change. In particular, the AD curve slopes upward. This will be explained intuitively in a moment, but mechanically the effect of a positively sloped AD curve is as follows:
Thus, when we consider only the supply-side effects of a tax cut, it has a negative impact on output and employment. Why is this?
Figure 5 clarifies the intuition for why labor tax cuts become contractionary at zero interest rates while being expansionary under normal circumstances. The key is aggregate demand. At positive interest rates the AD curve is downward-sloping in inflation. The reason is that as inflation decreases, the central bank will cut the nominal interest rate more than 1 to 1 with inflation..., which is the Taylor principle... Similarly, if inflation increases, the central bank will increase the nominal interest rate more than 1 to 1 with inflation, thus causing an output contraction with higher inflation. As a consequence, the real interest rate will decrease with deflationary pressures and expanding output, because any reduction in inflation will be met by a more than proportional change in the nominal interest rate. This, however, is no longer the case at zero interest rates, because interest rates can no longer be cut. This means that the central bank will no longer be able to offset deflationary pressures with aggressive interest rate cuts, shifting the AD curve from downward-sloping to upward-sloping in (YL,πL) space...
The reason is that lower inflation will now mean a higher real rate, because the reduction in inflation can no longer be offset by interest rate cuts. Similarly, an increase in inflation is now expansionary because the increase in inflation will no longer be offset by an increase in the nominal interest rate; hence, higher inflation implies lower real interest rates and thus higher demand.
Once again, however, demand side effects are missing. Tacking those on gives:
Thus, the overall effect on employment depends upon the net effect of the AD and AS shifts. If the AD shift dominates, as I suspect it would, this policy will still have positive effects on output and employment. But the size of the effect depends upon the strength of the demand side shift, and how strong the shift would be is an open question, particularly given the degree of household balance sheet rebuilding we are seeing which causes the tax cuts to be saved rather than spent. [The timing matters as well with the AD effects generally coming first, so in the SR the demand side effects should dominate. If so, that is a reason to be a bit more supportive of these policies.]
Another way to think about this is the following. Supply is not the problem right now, it's lack of demand, and a policy that encourages more supply and threatens deflation is not helpful except to the extent that it increases aggregate demand in the process. Other types of policies can avoid this problem, see, for example the sales tax cut discussed on page 20 or the discussion of fiscal policy multipliers on page 17, but they may not have the same political feasibility as tax cut for labor, which itself doesn't seem all that likely give the degree of opposition it will likely hit in Congress (the sales tax cut would be difficult to implement given that sales taxes are levied at the state level, and there's no chance that government spending increases will pass Congress right now; on the politics of a payroll tax cut, see the end of this post).
[Note: The demand-side effects were left out of the paper to keep the mathematics tractable, and it may be that simply tacking on the demand-side effects as I've done (the red lines) isn't quite correct. I think it's okay, but if anyone can speak to this, that would be great. Also, the policy analyzed in the paper is best interpreted as a payroll tax cut on the worker side. I don't think it matters if the cut is on the employer side, and I hope the administration doesn't pursue this anyway since the employer side tax cut may not pass through to labor fully, or much at all in the very short-run, but, again, if that matters and someone can speak to this point, please do.]