I have sympathy with arguments for flat tax rates and for progressive tax rates, but I think the best arguments against progressive tax rates are the declining marginal utility of government spending, and the associated poor allocation of resources. Progressive taxation feeds on the desire to tax individuals more responsibility, where those that have more, put more back into society. Just as we’d ask a man in a desert with ten thousand water bottles to give more water to thirsty people than a man with five.
The idea being, if a proportionate sum of money has far less marginal value to super rich George, and more to others where it can do more societal good, then there is a reasonable call for George to wish to be able to contribute to wider societal needs. It’s not difficult to demonstrate a utility function with declining marginal utility for which the utility of the higher parts of someone’s income provides less utility, and taxation provides higher overall utility. But only up to a point. When you have a case, as you do in the UK, where declining marginal utility of government spending comes into play, and additional taxation leads to misallocation, waste, bureaucratic overhead, and spending on things that most citizens do not want, any theoretical gains from redistribution are eroded by poor execution in practice.
So while the logarithmic utility function captures this idea of diminishing marginal utility - that the more you earn, the less each extra pound matters in terms of increasing utility – a tax system that tries to ensure that, after taxes, the marginal utility is more equal for everyone, is likely to be better under a flat tax system if the private allocation of resources has much more utility for the poorest than the public allocation.