The Prime Minister gave a speech on welfare today in which he gives various examples of the 'mess of perverse incentives, mind-numbing complexity and real unfairness' bequeathed to the coalition by the previous government. I'm interested in this one:

'Take a couple living outside London.He’s a hospital porter, she’s a care-worker. They’re both working full-time and together they take home £24,000 after tax. They’d love to start having children – and they know they’d get some help from the state if they did so. But with the mortgage and the bills to pay, they feel they should keep saving up for a few more years.

'But the couple down the road, who have four children, haven’t worked for a number of years. Each week they get £112 in income support, £61 in child benefit, £217 in tax credits and £141 in housing benefit – more than £27,000 a year. Even after the £26,000 benefit cap is introduced, they’ll still take home more than their neighbours who go out to work every day. Can we really say that’s fair?'

£24k in work vs. £27k out of work: how could any government allow such unfairness? But there is another way of looking at this.

Let's change three details in the story. Let's say the couple with four kids has been working up until very recently- both partners have lost their jobs. This doesn't affect their entitlement in any way. As the PM says they could be entitled, depending on their rent level and the age of their children, to £27,000 a year (with different assumptions, they could be entitled to more).

Now let's give our working family four kids so that we are comparing families which are identical in all relevant respects other than employment status.

Finally let's move things back in time so that we are looking at the situation the coalition inherited rather than the situation as it stands now: after all there have been changes to a number of important variables since 2010. I'll admit there's another motivation for going back to 2010 though: that's when the DWP stopped updating its published tax benefit model, and without a model it's very difficult to compare all the components of income.

Working from the 2010 tax benefit model and minimum wage rate, our couple where both are working full-time have take-home pay of £21,700 rather than the PM's £24,000. This isn't their income however. They are also entitled to in-work benefits. How much depends on whether they face childcare costs: as they have four children and they are both working full-time, that doesn't seem unlikely. They're entitled to a maximum of 80% of costs of up to £200 a week (this is 2010, remember). If they are receiving working tax credit to help with childcare, this affects how rapidly child tax credit is withdrawn as income rises. Assuming they face childcare costs of £200 a week, they will have the same entitlement to child tax credit as the non-working family, as well of course as the same child benefit entitlement. They are also eligible for housing benefit and a small amount of council tax benefit. Summing all of these elements, they would be entitled to a total of £21,000 of in-work benefits.

So the working family is getting quite a lot of benefit, and of course they have their take-home pay as well. The result is that their total income Before Housing Costs - that is, on the same basis as in the PM's examples- is £43,000. Our non-working family, using the 2010 model, would have had an income of £27,500.

Is this comparison reasonable? I could be accused of just making up the scenario, but then so could the prime minister: neither of us has done any checking to see if these figures correspond to anything happening in the real world. But looking at things from the PM's point of view, where it doesn't matter if large chunks of benefit income are going on housing or childcare costs, I am struggling to see what his comparison is supposed to illustrate. Who should the minimum wage working couple with no children resent more: the recently unemployed couple whose income is about £5,000 more than theirs because they have four children and for no other reason: or the other minimum wage working couple whose income - on the PM's definition- is £21,000 more than theirs, again because they have four children?

The PM's claims turn on the fact that he is comparing families with children with families without children, not on the fact that one is working and one isn't. This is one way of conjuring up the sense of 'real unfairness' he is talking about. Previously the government used a different method, seeking to foster grievances by comparing some of the income of working families with all of the income of non-working families, as noted here . In today's example, all of the income is brought into the comparison on both sides, but the families are of different sizes. The comparison is no less misleading.

If promoting a policy turns on making deceptive comparisons, the policy is unlikely to be a good one. Of course it's possible the PM, who is said to have a weak grasp of detail, is unaware of the con-trick he performed in today's speech. It would not be difficult to set the record straight. Indeed, fairness, to voters as well as to out-of-work families, would seem to demand nothing less.

(You can reproduce these calculations using the model yourself I've assumed the kids are all under 5 and I've set the rent level at £141 to match the PM's example. Bear in mind that a couple with two earners also has two tax/National Insurance contributions allowances so their take-home pay will typically be higher than a single-earner couple with identical gross earnings. Work out take home pay for one person full-time at minimum wage first using the model, double it and then use this figure to find benefit entitlement for that level of take home pay in the model.)