Some questions are bad questions. This is one of them. We can get a clearer and more useful answer if we change the question. We can avoid wasting a lot of time arguing at cross purposes.
Here's a better question: "If we used fiscal policy instead of monetary policy to remove a shortage of aggregate demand, would that switch from one policy to another have distributional consequences?" The answer is almost certainly yes, though what those distributional consequences are will depend on the exact nature of the fiscal policy.
Here's an even better question: "If we assume that monetary policy always responds to changes in fiscal policy to keep aggregate demand at the same level, will changes in fiscal policy have distributional consequences?" Now the answer is certainly yes, though what those distributional consequences are will depend on the exact nature of the fiscal policy.
[Update: now I think about it, I think my third question is actually the same as my second question; just a clearer way of saying it.]
It all comes down to the "assignment question": what policy instrument is assigned to which target? Fiscal policy is multi-faceted, because there are lots of different types of government expenditures and lots of different types of taxes. Monetary policy isn't. You need lots of numbers to describe fiscal policy, and only one number to describe monetary policy. Fiscal policy can in principle be used to change the distribution of income/wealth in almost any way you want; monetary policy can't. It makes sense to assign income distribution to fiscal policy and aggregate demand to monetary policy. If you don't like the distribution of income, then change fiscal policy, and let monetary policy adjust in response to keep aggregate demand at the right level.
This post is in response to many people I have heard asking the title question and arguing against using monetary policy to increase aggregate demand because of the distributional consequences. But most recently it is in response to Ashwin's post in Macroeconomic Resilience.
"A quantitative easing program focused on purchasing private sector assets is essentially a fiscal program in monetary disguise and is not even remotely neutral in its impact on income distribution and economic activity. Even if the central bank buys a broad index of bonds or equities, such a program is by definition a transfer of wealth towards asset-holders and regressive in nature (financial assets are largely held by the rich). The very act of making private sector assets “safe” is a transfer of wealth from the taxpayer to some of the richest people in our society. The explicit nature of the central banks’ stabilisation commitment means that the rent extracted from the commitment increases over time as more and more economic actors align their portfolios to hold the stabilised and protected assets."
There's a lot I would disagree with in that paragraph.
First, it is the central bank's threat to buy an indefinitely large amount of an indefinitely wide variety of assets for an indefinitely long time, conditional on NGDP (or whatever) being below target, that does the job of increasing expected and hence actual aggregate demand. Not the actual purchases.
Second, the large current size of central banks' balance sheets, and the high prices of the assets held by central banks, is a consequence of tight monetary policy, not loose. A credible threat to buy more assets conditional on NGDP being below (a sensible) target would mean central banks would have to shrink their balance sheets by selling assets, and would cause the prices of those assets to fall. Low nominal interest rates are a consequence of low NGDP expectations.
Third, monetary policy works not by changing central banks' (current and expected future) stocks of assets but by changing central banks' (current and expected future) stocks of liabilities. Anybody can buy, sell, and hold those assets; only central banks (and counterfeiters) can issue high powered money. Central banks are small players in the market for all other assets, but are very big players indeed in the market for high powered money.
But mostly, I think Ashwin is asking the wrong question. I can't stop people asking the question "would loosening monetary policy to increase aggregate demand cause the distribution of income/wealth to worsen?". My guess is that some people would indeed be made worse off, like people who hold a high proportion of their wealth in safe nominal bonds (though I'm not sure if that counts as a worsening of inequality or not). But I can't imagine that anyone would use that as a reason for not increasing aggregate demand. After all, those who would probably benefit the most would probably be those with the highest risk of unemployment, who are likely to be among the most poor. I hope nobody would argue "we just have to live with the recession, because if we cured it some rich people would benefit".
It all comes back to the question of the assignment of targets to instruments. Like most economists for the last 40 years, I would assign aggregate demand to monetary policy and the distribution of income to fiscal policy. You use monetary policy to target aggregate demand, because that's about the only job it's good at, and you don't want to ignore an instrument when you don't have enough instruments to leave one idle. And you use fiscal policy to target income distribution, because it's good at that job and monetary policy is bad at that job. Then ask: "if monetary policy responds to fiscal policy (and everything else) and keeps aggregate demand where it should be, will fiscal policy changes have distributional consequences?" The answer is: "of course they will". It's just standard micro public finance.