One question advocates of expansionary fiscal policy in situations like the global economy's current configuration must face is Mankiw and Weinzerl's  question: why not just use monetary policy instead?
The principal argument for monetary policy is that, by modifying asset supplies and thus asset prices, it induces households and businesses to boost their spending on things that they almost bought anyway. Thus--for marginal policy shifts, starting out at a first-best optimum, and if the relative distribution of wealth corresponds to social welfare (or if questions of the relative distribution of wealth are left to a more openly political process and walled-off from technocratic macroeconomic questions of stabilization policy)--monetary policy will not push you far away from the free-market optimum.
Fiscal policy, by contrast, works through expanded government purchases ΔG. These must be financed by distortionary taxes to amortize the debt in the future. These taxes do drive a wedge between the social and the private values of output in the future. And what the government buys is determined by a political rather than by an optimizing economic logic. 
In such a setup, the conclusion of Mankiw and Weinzerl that monetary policy has the exclusive role to play is straightforward: One stabilization policy tool--monetary policy--is non-distortionary. The other stabilization policy tool--fiscal policy--is distortionary. If monetary policy can do the job, there is then no need for fiscal policy. And if you do resort to fiscal policy, use the fiscal policy that is most effective at getting people to spend money on the things they were at the tipping point of buying anyway--use the investment tax credit rather than direct government purchases or tax cuts which might well not be spent. End of argument.
But are the assumptions correct? Can monetary policy do the job?
There is little  doubt that it can do the job--and that the conclusion is sound--in normal times, when the short-term safe nominal interest rate is away from its zero nominal lower bound, and when small bond sales to shrink and bond purchases to grow commercial-bank reserve deposits shake the entire intertemporal price structure. 
But does the same hold true in a liquidity trap, when short-term safe interest rates are at their zero nominal lower bound? At the zero nominal lower bound monetary policy as stabilization policy has two potential modes of effectiveness:
Even with short-term safe nominal bonds at par, the market flooded with excess reserves, and thus with reserve deposits perfect substitutes for short-term Treasuries, the monetary authority can take duration and default risk onto its balance sheet and thus free-up risk-bearing capacity to improve borrower access to credit.
Even with short-term safe nominal bonds at par, the market flooded with excess reserves, and thus with reserve deposits perfect substitutes for short-term Treasuries right now, the monetary authority can promise it will keep interest rates lower and inflation rates higher in the future than its standard reaction function would warrant.
The first of these requires enormous gross asset swaps in order to free up any noticeable amount of risk-bearing capacity. Moreover, it requires that the monetary authority have the power and the will to commit the taxpayers to backstopping it and bearing substantial amounts of default risk.
The second faces all of the problems of the dynamic consistency literature, but in reverse. After the economy has recovered, what reason is there for those at the head of the monetary authority to deviate from their standard policy rule, as their predecessors who are in office now promised they would? And since those currently at the head of the monetary authority cannot bind their successors, why should their promises today be more than cheap talk?
There are costs of letting inflation in the future rise above the monetary authority's target level. First, there are the consequences for the monetary authority's credibility as an inflation fighter. Second, there are a variety of costs arising from the tilting of the duration structure of the debt that comes from monetary authority decisions to oppose a current downturn by shifting to a higher target for the price level in the future.
Whenever there are problems of monetary policy commitment, the doctrine of fiscal dominance suggests that expansionary fiscal policy can help resolve them. The larger the debt, the larger the absolute amount of debt amortization that an optimizing central bank will perform via seigniorage, and the higher the price level and the inflation rate.  Fiscal dominance--having the fiscal authority set current borrowing and future primary surpluses and requiring the monetary authority to print money to fill in the gap--can make fiscal policy extraordinarily powerful. For example, consider the model of Cochrane (2010) , in which a 1% increase in the national debt leads to a 1% increase in nominal spending not just this year but in all future years. If monetary expansion via quantitative easing is not credible as a commitment to looser future monetary policy because the easing transactions can be easily unwound, fiscal expansion funded by money-printing is a credible commitment to looser future monetary policy because there is no easy way for the government in the future to sell off its bridges and dispossess its citizens of the human capital bought and thus to unwind the monetary expansion.
Moreover, the claim that monetary policy is non-distortionary is subject to question when the real interest rate that produces total spending equal to potential output is negative. In such a case, monetary policy works by artificially pushing the real interest rate r down well below its full-optimum level to compensate for other market failures. The relative overproduction of long-duration and underproduction of short-duration assets in the present is not small, and the claim of no first-order losses is not clearly relevant.
In addition, there are the behavioral-finance fears: a market in which interest rates are very low is a market in which many financial institutions will make inappropriate judgments about long-run risk and return, as they find themselves driven by institutional and other imperatives to “reach for yield”.  A low cost of funds makes mark-to-model accounting easier to sustain. When the cost of funds is substantial, maintaining positive cash flow requires much more frequent testing of models by marking-to-market. And an entire financial industry engaged in unchecked mark-to-model accounting is dangerous.
It is important to get the overall level of production right--to match total spending to potential output. It is also presumably important to direct spending toward high-value commodities. It is important to get the balance between private and public purchases right. And it is important to get the balance between short-duration and long-duration assets right.
Thus fiscal and monetary policy are likely to both have proper stabilization policy roles to play. 
 Gregory Mankiw and Matthew Weinzerl (2011), "An Exploration of Optimal Stabilization Policy", Brookings Papers on Economic Activity 2011 (Spring) http://www.brookings.edu/~/media/Files/Programs/ES/BPEA/2011_spring_bpea_papers/2011a_bpea_mankiw.pdf
 Of course, these arguments cut the other way if rent-seeking and problems of collective action lead to underprovision of public goods by the government. Then every excuse to boost government purchases of public goods should be seized, and fiscal expansion dominates expansionary monetary policy.
 Alan Blinder (2004), “The Case Against the Case Against Discretionary Fiscal Policy” (Princeton, NJ: Princeton University).
 Why conventional open-market operations are so effective in shaking the entire intertemporal price structure in normal times is a question with serious depth.
 Olivier Blanchard (2005), “Fiscal Dominance and Inflation Targeting: Lessons from Brazil”, in Fran-cesco Giavezzi, Ilan Goldfajn, and Santiago Herrera, eds., Inflation Targeting, Debt, and the Brazilian Experience, 1999 to 2003 Cambridge: MIT Press).
 John H. Cochrane (2010), "Understanding Policy in the Great Recession: Some Unpleasant Fiscal Arithmetic" http://faculty.chicagobooth.edu/john.cochrane/research/Papers/understanding_policy.pdf
 Charles Kindleberger (1978), Manias, Panics, and Crashes (New York: Basic Books): "Too low an interest rate is a special case of what is perhaps a wider phenomenon--the pricing of financial innovations…. [I]nnovations may be underpriced as loss leaders so they will be more generally accepted…. [U]ndue risks may be taken by recent entrants in an industry as they reduce their selling prices to increase their market shares. One noticeable example is that of Jay Cooke, the last prominent banker of the early 1870s to back a railroad…"
 This conclusion is reinforced by considering the consequences of model uncertainty. See William Brainard (1967), "Uncertainty and the Effectiveness of Policy", American Economic Review http://cowles.econ.yale.edu/P/cp/p02b/p0257.pdf