... estimate how many smokers there are and multiply it by the estimate of how many cigarettes they each smoke. They then subtract the number of recorded sales - plus an estimate of how much is bought legally abroad - from the total and the figure that emerges is the amount of non-duty paid tobacco.
I have suggested that the decline in legal tobacco sales since 2009/10 - when tobacco taxes began to rise well above inflation - has been far greater than the decline in smoking prevalence would predict. In 2009/10, 49 billion cigarettes were sold. By 2014/15, this had fallen to 33 billion, meaning that sales had fallen by a third. Sales of hand-rolling tobacco, meanwhile, had remained the same at around 6.6 million kilograms. The graphs below show legal sales of cigarettes and other tobacco (mainly hand-rolling tobacco) since 1992. Note the big dip in 2000 when cross-border shopping from the EU began in earnest.
Over this period, the smoking rate has fallen by just 10 per cent (or two percentage points), from 21 per cent to 19 per cent, and half of that drop has taken place since 2010. Since 2010, the smoking rate has fallen by just five per cent but cigarette sales have fallen by more than a quarter. Moreover, the population has grown by four per cent.
It is possible that this can be explained by smokers sharply cutting down their consumption (perhaps because of e-cigarettes), but it is also possible that higher prices have pushed more people into the illicit market.
Or it could be a bit of both. Unfortunately, the HMRC estimates are so rough and ready that it is impossible to tell. Last time I wrote about this, HMRC figures showed a rise in the illicit market share since 2011/12 (from 7 per cent to 10 per cent in 2014/15), but no real change since 2009/10 when the estimate was 11 per cent.
It should be noted that any figure between 4 per cent and 12 per cent is possible in any of these years. Look at the confidence intervals. This is a broad guesstimate that depends on a lot of assumptions.
The importance of assumptions can be illustrated by looking at the most recent HMRC estimates which use an updated methodology. All the figures have suddenly changed. Whereas HMRC previously found no change between 2013/14 and 2014/15, they now find a four percentage point decline - from 11 per cent to 7 per cent - followed by a near-doubling of the rate, from 7 per cent to 13 per cent.
HMRC say that their new methodology takes into account evidence about people who 'falsely deny they smoke'. Rightly so, but estimating how many smokers claim to be non-smokers also involves a large element of conjecture. Does the new methodology get us any nearer the truth? It is hard to say. What is clear is that the figures are highly sensitive to changes in assumptions.
HMRC add their estimate of the illicit market to the legal sales data and conclude that the total number of cigarettes sold in Britain (legal and illegal combined) has fallen from 56 billion in 2009/10 to 38.5 billion. This is a drop of nearly a third (31 per cent) in a period when smoking prevalence has declined by just ten per cent and the population has grown by four per cent. And so the puzzle remains.
A reduction in the number of cigarettes smoked per person offers a partial explanation. The ONS says that average daily cigarette consumption in England fell by two percentage points between 2009 and 2012 (from 14 to 12 for men and from 13 to 11 for women). Its most recent dataset shows a figure of 11 per day (both sexes combined). This suggests an overall decline in cigarette-per-smoker consumption of 18.5 per cent since 2009.
This is a substantial drop but not enough to explain the overall decline in (legal) sales. There were 49 billion cigarettes sold legally in 2009/10. Reducing this by 10 per cent to account for the decline in smoking prevalence takes it down to 44.1 billion.
Reducing that 44.1 billion by 18.5 per cent to take into account lower consumption by smokers gives us 36 billion.
Increasing that 36 billion by four per cent to take into account population growth gives us 37.4 billion.
But HMRC records legal sales of 31 billion in 2015/16.* HMRC reckon that a further 1.5 billion cigarettes were brought in from the EU legally. This takes our 37.4 billion down to 35.9 billion, so we are still five billion cigarettes short of what we might expect, but this could be partially explained by a shift to rolling tobacco. Despite the decline in smoking prevalence, sales of hand-rolling tobacco remain the same as they were in 2009/10.
This is a rough and ready calculation, but it suggests that HMRC's estimate that the illicit trade makes up between 8 per cent and 17 per cent of the cigarette market can be justified, although the real figure seems closer to the high end than the low end. HMRC's mid-point estimate of 7 per cent for the previous year certainly looks implausibly low.
Even if HMRC's mid-point estimate of 13 per cent is correct, the number of illegal cigarettes on the UK market is mind-boggling. It suggests that six billion cigarettes were sold on the black market in 2015/16, along with ten million kilograms of rolling tobacco. That's in addition to 1.5 billion cigarettes and 3.2 million kilograms of rolling tobacco that avoid UK duty by being brought in legally from the EU.
And, to repeat, HMRC's estimates are incredibly sensitive to changes in assumptions. For example, if the average smoker consumes 11.5 cigarettes a day, as opposed to 11.0 cigarettes as assumed above, it leaves another 1.5 billion cigarettes unaccounted for.
We are talking about a serious problem for the government and yet there does not seem to be any serious effort being made to establish the facts. For a relatively small outlay, the government could conduct empty pack surveys and get a better idea of what is going on. Unfortunately, there are perverse incentives at work because some people would rather live in darkness than light a candle. The customs agencies want to be seen to have got a grip on tobacco smuggling and the government (along with anti-smoking groups) wants to believe that it can raise taxes without fueling the black market.
Three things are clear, however. Firstly, that tobacco taxes hit the top of the Laffer Curve in 2012/13 when £9,681 million was raised. Revenues have fallen every year since and stood at £9,485 million in 2015/16, despite higher rates of duty.
Secondly, that whichever methodology is used, the illicit cigarette market has grown in recent years as a share of the overall market. The 13 per cent mid-point estimate in the most recent dataset is higher than it has been for a long time.
Thirdly, that there are reasons to think that even 13 per cent could be a significant under-estimate, but until efforts are made to measure the illicit market properly, no one will know for sure.
* The tobacco tax gap documents says 32 billion, but the tobacco bulletin says 31 billion (30,974 million, to be precise).