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You're probably right about inflation. Nevertheless, there are a number of other factors that could mitigate the growth effect due to spending including inflation, inefficiencies in the market, inefficiencies in the purchase choices made by consumers, purchasing foreign goods, and so on.
Inflation is a hurdle for any stimulus plan. As a specific criticism of tax cuts, its a weak argument.
Tax cuts that target consumer spending are fairly simplistic and generally aren't plagued by the market inefficiencies of investing.
I have a hard time envisioning a scenario where a consumer willingly makes an inefficient purchase. Technically, any transaction where both parties end up happier is a bridge towards efficiency. E.G: I have $100, and a producer has a fur coat. I'm happier with the coat than the $100, and he's happier with the $100 than the coat. It doesn't matter that coat has no productive value, everyone is better off.
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This is true. Savings does equal investment though not all investment grows the US economy; some could be foreign. I was trying to emphasize debt reversal though. Paying off debt doesn't grow the economy (at least not how it's currently calculated).
These are little snippets at something thats true overall. Certainly, people can pay off debt with their tax savings, but its unlikely that all of the tax cut will go towards paying off debt. Alot of it will ultimately be used to buy stuff, which means that, holding everything else constant (government spending), a tax cut will provide more growth than no tax cut.
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I agree that tax cuts will have a broader impact across the economic spectrum, but this doesn't mean the net economic impact between consumer and government spending will be different. Breadth-based and depth-based spending can have an equivalent impact.
I think your other two arguments are off mark except this one. If government spending increases growth, and tax cuts increase growth, will reducing government spending to increase tax cuts result in less, more or no growth?
From a theoretical perspective, theres little reason to assign much difference between government and consumer spending. GDP does not care whether Tim buys a pizza or Uncle Sam does, so much as it gets bought it contributes towards GDP and is a factor in growth.
But from a practical perspective, there are several reasons why government spending is not the way to go. First, as I mentioned before, tax cuts have a broader impact on the economy as a whole. Government cannot spend money at all stores everywhere, but consumers might be able to. In the long run, this won't make much of a difference because there should be a chain effect where people pass money for products amongst themselves until all money and products find the right places. In the short run, however, more people will feel relief from a tax cut than government spending.
The second reason is that government spending is not always efficient (thats an understatement!). I chuckled a bit when I read your comment "inefficiencies in the purchase choices of consumers." Consumer spending looks like a utopia of efficiency compared to government spending.
Part of the reason is actually pretty practical: its a hard job to tax people and then guess at a transaction that makes people collectively better off than the tax dollars they spent. To take the fur coat example: if a politician was authorized to tax $100 from the people and spend it on a coat, people might end up wanting different coats. Its hard to choose the best coat given the diversity of opinions.
The other part is that government officials often use tax money to satisfy their own desires. In fact, I will argue that government is more likely to make inefficient transactions than private citizens
because elected officials do not personally bear the costs of the transaction . If a normal person were given a tax rebate for $100, and they saw a coat that to them was only worth $50, they would, efficiently, choose not to buy the coat. On the other hand, if a government official saw the $100, and it was only worth $50 to him, and he was allowed to spend $100 of tax payers money on the coat, he would end up making the purchase because the coat effectively costs him nothing. This example is a little unfair, because, realistically, government officials don't have authority to spend money willy nilly on personal items like I described. But the lines of public and private expenditures often get blurred, and I have great confidence that everyone on AD can extrapolate my example towards cronyism, bridges to nowhere or other pointless government projects. The overarching idea is that an elected official might make spend $100 of taxpayer money for $50 of personal benefit.
Whether inefficient government spending is due to practical difficulties or corruption, a tax cut which cut these inefficient programs, and then generated efficient consumer spending, would help the economy overall in the long run. Money given back to the people is better spent than building national parks or pointlessly invading countries. I can think of a couple things I would like to cut

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As a practical matter, the government, when employing Keynesian economics, generally funds tax cuts through borrowing anyway while they ramp up government spending. Those cuts don't suffer the conflicting nature of reducing government spending.
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The model they employ may be powerful and useful but it has not been proven. Sorry to be such a stickler on this. And again, I'm not arguing that economic growth cannot occur; I am saying that this hasn't been proven.
There is rarely such strict empirical proof available for all social studies, and particularly ones with such a broad claim.
The study I linked used real US data from a reliable economist at a reliable source: it demonstrated that tax cuts are partially self-financing.
This is the first time I've gone up against a "non-position" ("I'm not arguing that economic growth cannot occur; I am saying that this hasn't been proven"), and I feel you have a right to an explanation of the logic I'm employing, and a source of reasonable data. However, I've bent over backwards, particularly with this last post, to provide both. I don't feel I have to jump through a specific hoop in order to be confident that tax cuts increase growth.
After all, as the study I linked says:
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The consensus view is that tax cuts indeed influence national income, but not to the extent that they are fully self-financing.
Even Paul Krugman agrees the last round of cuts were at least a little stimulative. To be blunt,
Ted and I are in good company.