News coverage of important and complicated economic developments is often very helpful, but sometimes it leaves more questions that it answers. I was trying to get to grips with the Budget's implications for welfare this morning, but the few headline measures don't really mean anything if you don't understand how tax credits work. So this morning I spent a few minutes getting to understand the system and a few more minutes trying to model the reforms.

There are two main sorts of tax credits in the UK: Working Tax Credits, paid to those in work but on low incomes, and Child Tax Credits, paid to those with children. The two credits replaced the old Working Families Tax Credits, which basically did the same but in one go.

There is a maximum value for each component of each credit, which is reduced as your income increases. There is what is called the first threshold (presently, £6420), where earning below this means you get the maximum. At present, tax credits are taken away at a rate 39p for each pound earnt over that threshold. They are technically ordered as well, so that the first credits to be removed are working tax credits (which have a basic component, a component for being a lone parent or a couple, and a component for working over 30 hours a week), then childcare, then the variable components of Child Tax Credits (children, baby, disability) except for the family component (which is the basic fixed element). The family component remains at £545 until the second threshold, £50,000, and then removed at a rate of 6.7p for every pound earned after that.

OK so far? The key changes in yesterday's budget are that the rate of removal is being changed to 41p for each pound for all credits (including the family component of the Child Tax Credit) and that the second threshold is being reduced to £40,000 next year. The one change that hasn't really been picked up on is that the second threshold for the family componet will be scrapped in 2012-13, so the family component will be removed immediately after all other components are - I'll show an example in a minute, but it basically hits everyone earning from about £23000 to £40,000. The baby component is being withdrawn.

At the same time, the children component of the Child Tax credit is being increased by £150 per year (in real terms) and another £60 the following year. Currently, this component is £2300 per child. This shifts the benefits from tax credits in favour of the lowest earners. Added to this, the tax allowance rose by £1000. Anyone earning between the current level and the new level will now pay no tax, and anyone earning above it will pay about £200 less tax each year. Those who have been taken out of tax will gain between 20p (for the person earning £1 more than the old threshold) and £199.80 (for the person earning £1 less than the new threshold).

Ignoring the income tax issue for the moment, I wanted to figure out who is better or worse off from the tax credit changes, so I put all of this into a model and looked at changes for benefits for incomes ranging from £1000 per year to £70,000 per year. I just focused on families (single parent or otherwise) with children (not babies and not disabled), so they receive a working tax credit - maximum of £3810 if they earn below £9,000, £4,600 if they earn over that, because at that point you would expect the person works 30+ hours in a week and receives an extra £790 - a child tax credit (maximum of £2300 per child in 2010, £2450 in 2011 and £2510 in 2012) and the family component (maximum of £545). I ignored childcare credits and child benefit for simplicity and kept all other benefits and the first threshold constant.

If you have one child, you are better off in both years if you earn less than £13,000 per year, by up to £210. This is important for government aims about poverty - the benchmark threshold for poverty in the UK is around £13,000 (about 60% of median income). So any claims that this Budget do not hurt the poorest stand up to some scrutiny here.

Up to £17000, you gain in 2012-13, but lose less in 2011-12. Above that, the losses this year from the steeper withdrawl tend to outweigh the gains the following year from the higher children component. The total net losses are between £0 and £120, depending on income.

Families earning between £24,000 and £40,000 are not affected this year, as they only received the family component. Above £41,000, the family component disappears, so familes between £41,000 and £50,000 lose all or most of their current credits this year. Next year, families earning between £24,000 and £41,000 will lose the family component due to the scrapping of the second threshold. This is why it is surprising this reform hasn't been picked up on - it will affect a lot of people to the tune of £550 per year. At the same time, it's not surprising how little it has been mentioned, because it's a bit complicated to get at.

For example, if you have two children, the patterns are roughly the same, but the exact thresholds change, and the headline "Families to lose £550 per year if they earn a certain amount per year" wouldn't work well. In this case, it is families between £31,000 and £40,000 that will lose the family component of their tax credit in 2012-13.

One last thing is the income disregard change from £25,000 to £5000. This, unlike what I thought after my first reading, is not a threshold, but a way of dealing with unexpected changes in income. So you earn £12,000 the previous year and are given the appropriate tax credit payment for the coming year. At the end of the year you earn £14,000. The income disregard means you haven't got to pay any credit back if your income change is small enough. "Small enough" used to mean £25,000 a year, and will now mean £5,000 per year. A tightening of the belt, for sure, but not one anywhere near as scary as the magnitude of the numbers (a drop of 80%), once you understand what it is. And now you do too.

Anway, that's all the numbers. The welfare state is a topic which raises strong emotions, often at the expense of facts. I would want to do nothing about the emotions, but hopefully this will do something for the facts.

By the way, our tax and benefit system hurts my head.

UPDATE: Some attention now being paid to the middle income families issue.