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"A USD 1,000 donation from a donor in the highest tax bracket costs that donor only USD 604. The federal government kicks in the remaining USD 396 in the form of a reduction in taxes", writes Ray D. Madoff in a piece for the New York Times.

Similarly, Paul Vallely argues in an article for The Guardian, that, for a tax-deductible donation, "the highest-rate taxpayer would need to pay out only £55, because the state would provide the other £45", hence "taxpayers contribute part of the gift".

While I do understand, that the government loses money in tax incomes, I don't understand how this directly increases the tax burden on other tax payers (except for, in the long term, the government might raise taxes to make up for lost tax revenues).

Can anyone explain how this works?

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Well you actually answered this question yourself. It is because of this:

While I do understand, that the government loses money in tax incomes, I don't understand how this directly increases the tax burden on other tax payers (except for, in the long term, the government might raise taxes to make up for lost tax revenues).

To be more precise an alternative is also for government to spend less but I think that the articles are arguing that ceteris paribus keeping government spending fixed government will have to somehow eventually collect the resources from economy in one way or other. In a long run government has to respect its intertemporal budget constraint. If someone pays less, taking government spending as fixed, someone will have to pay for that shortfall either directly or indirectly eventually at some point.

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