Let's quickly define what "share" and "take" mean. Consider a pie. The "share" (S) is simply a fraction of the pie. The "take" (T) is the weight of the given piece of pie. Given a pie that weighs W, the equation is T = S⋅W. Simple enough.
Clearly, for T to increase, either S or W need to increase (or both). However, in the case of the American pie called GDP, there is also a relationship between S and W. As the governments' share (S) of GDP increases, the pie makers (the private sector) make a relatively smaller pie, so W decreases. Therefore it's not clear if the governments' take will go up our down if they choose to increase their share. If GDP decreases faster than S increases, the take would decrease. If GDP (W) decreases slower than S increase, the take T would increase.
Let's say for a moment that the governments wanted to exactly maximize its take. To do so they would pick the share that maximizes the equation T = S⋅W and therefore the change in W would exactly offset the change in S if S increases or decreases. Let's say the governments have successfully done so and that the current share of 1/3 maximizes their take. If so, we would predict that for each percent increase (or decrease) of the governments' share of GDP, GDP would decrease (or increase) by approximately 3%.
Sure enough, according to a recent paper by Christina and David Romer of the University of California, Berkeley, tax revenue increases are a significant negative for the economy and the relationship between share of GDP and size of GDP happens to exactly maximize the governments' take:
Our baseline specification suggests that an exogenous tax increase of one percent of GDP lowers real GDP by roughly three percent.Governments exist only to exert and extend their power via a parasitic relationship with their economic host. One-third GDP maximizes our governments' take so that's what they take. No more, no less.