The bottom line is that:

"real median US household income has risen by about 30 per cent." and "US GDP per person has more than doubled"

mathematically, would have to be explained 100% by:

"Average household size has fallen from around 3 to 2.6"

and

"inequality as measured by the ratio of mean to median household income"

if the following are true:

-- Assumption 1) You get rid of the "around"s and "more than"s and plug in the precise numbers. For example, you plug in the exact amount real median US household income has risen, rather than "about 30 per cent".

-- Assumption 2) You assume the income used in "GDP per Person" is the same as the income used in "Average Household Income". In other words:

a) GDP per Person = Total National Income/Number of People

b) Average Household Income = Total National Income/Number of Households

As long as Total National Income is measured the same for both of these, as long as it's the same number, then you're fine. If not, then this is a source of your problem, and a great lead. You should look at how these are measured differently.

-- Assumption 3) There's no measurement errors, math errors, or other mistakes.

Now, if all of this is true, then the following is true:

First, let's write what you want to explain in mathematical notation:

"real median US household income has risen by about 30 per cent." and " US GDP per person has more than doubled"

Can be written mathematically as:

(M1/M0) / [(I1/P1)/(I0/P0)] ; Let's call this Term 1,

Where:

Mi = The median household income at time i

Ii = Total National Income at time i

Pi = The number of people in the country at time i

Now, what are you trying to explain this Term 1 with?

a) "Average household size has fallen from around 3 to 2.6", this is the gross change in average household size. Mathematically, it can be written as:

(N1/N0)

where Ni = The average number of people in a household at time i.

b) "inequality as measured by the ratio of mean to median household income". Mathematically that can be written as:

[(I1/N1)/M1] / [(I0/N0)/M0] ; Let's call this Term 2,

where,

Ii = Total National Income at time i.

Ni = The average number of people in a household at time i.

Mi = The median household income at time i

Next, if you do some algebra, you will see that:

Term 1 = (N1/N0) / (Term 2) ; let's call this Equation 1.

which shows that what you are trying to explain, Term 1, can, in fact, be completely explained, 100%, by the things you said couldn't completely explain it, and were wondering why, namely (N1/N0) and Term 2. These can, in fact, explain it 100%, but only as long as the assumptions 1-3 that I've made above are correct. So you want to look for problems in these assumptions.

Now, let's put our relationship equation, Equation 1, into words, to make things clearer:

Equation 1 means:

(The Gross Percent Change in Median Household Income) / (The Gross Percent Change in GDP Per Person)

=

(The Gross Percent Change in Household Size) / (The Gross Percent Change in the Ratio of Average Household Income to Median Household Income).

So, what's the effect if the Household Size stays the same -- meaning The Gross Percent Change in Household Size = 1, but the denominator doubles, that is The Gross Percent Change in the Ratio of Average Household Income to Median Household Income = 2?

If that happens, then the left hand side of equation 3 is cut in half; GDP Per Person grows twice as much as Median Household Income.

So, bottom line, what you have should explain everything, but only as long as assumptions 1-3 are true. The key assumption I'd look at is 2, the assumption that the income used in "GDP per Person" is the same as the income used in "Average Household Income". For example, they may not be including a lot of transfer payments in household income, or they may not be including all benefits, bonuses, and dividends and other capital income. And then there's income that people hide for reasons like tax evasion and other illegal activities.

Consider, UCLA economist Emmanuel Saez's March, 2008 paper, "Striking it Richer: The Evolution of Top Incomes in the United States (Update using 2006 preliminary estimates)" . He writes:

We define income as the sum of all income components reported on tax returns (wages and salaries, pensions received, profits from businesses, capital income such as dividends, interest, or rents, and realized capital gains) before individual income taxes. We exclude government transfers such as Social Security retirement benefits or unemployment compensation benefits from our income definition. Therefore, our income measure is defined as market income before individual income taxes. (page 1)

I don't know exactly what data sources you're using, but the answer to your puzzle certainly may lie in the details of how they are compiled.

Another point I'd like to add is that for income inequality, looking only at how quintiles have done is really misleading, because there has been so much increased inequality within the top quintile. It's in the top 1% and the top 1/10^{th} of 1% that you really see an explosion in income inequality.

Saez's data series' (available at http://elsa.berkeley.edu/~saez/) show that in 1978 the top 1% received 8 times the average income; by 2006, this soared to 23 times. For the top .01% the increase was from 86 times the average income to 546 times! And on top of this, Republican tax cuts starting in the Reagan administration and continuing through Bush II have made taxes far less progressive. The top federal income tax rate was cut by 35 points between 1979 and 2006. Average tax rates on the richest 0.01% were cut in half between 1970 and 2006, while taxes on the middle class were increased (From Princeton economist Paul Kurgan's most recent book, "The Conscience of a Liberal" (page 145) and his 2007 column, "Gilded Once More"). Another great source is the Economic Policy Institute's annual book, "The State of Working America".

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