For @michael_nielsen a "short" explanation of why (TFP growth) = (real GDP growth) - 0.7*(labor growth) - 0.3(capital growth)

Composing as a thread rather than a set of replies, so I can go back and edit if needed.

Assume we have a general aggregate production function linking GDP (measured in real output) at time t (denoted Y(t)) to capital and labor at time t via a function f(.) and TFP at time t (denoted A(t)):

Y(t) = A(t) f(K(t),L(t))

Take time derivatives and get:

dY/dt = dA/dt * f(K(t),L(t)) + A(t) {df/dK * dK/dt + df/dL * dL/dt}

We're going to move some stuff around now and I'm going to try and describe it on twitter.

Divide both sides by Y.
Left-hand side is now "real GDP growth"
After setting Y = A(t)f(K(t),L(t)) and canceling f(K(t),L(t)), first term on right is "TFP growth"

For the second term on the right, multiply by K/K to get:
A(t) * df/dK * K * (growth rate of K) / Y.

Here's the trick: A(t) * df/dK is the marginal product of capital - how much additional output you get from a little more capital. Denote this MPK.

In a competitive market, firms hire capital until the marginal revenue generated by capital equals the rental price (r) of capital: pMPK = r

Thus MPK = r/p and A(t) * df/dK * K / Y = rK/pY

rK/pY is total payments to capital, as a share of GDP.

Data tells us that number is 0.3

We can do the same tricks with the last term to get A(t) * df/dL * L * (growth rate of L) / Y = wL/pY

wL/pY is total payments to labor, as a share of GDP.

Data tells us that number is 0.7.

So now we have:

(real GDP growth) = (TFP growth) + 0.3*(capital growth) + 0.7*(labor growth)

We can rearrange this to derive the growth rate of TFP as indicated in the original equation.

Lastly, the assumption that you have competitive markets can be relaxed, but things get more complicated. This is getting beyond my area but my sense is that additional complexity does not fundamentally change the rule of thumb.

Did I get that right @DietzVollrath?

PS - @DietzVollrath's blog is a great explainer of other TFP issues. growthecon.com/blog/