Public Policy Sources #37:
Appendix 1 The algebra of growth accounting

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Recall that we start with the assumption that the production function for the economy takes the following Cobb-Douglas form:

Y(t) = A(t)K(t)x L(t)1-x (A1)

We also assume that 0 < x < 1. To perform growth accounting, we do the following. First, take the natural logarithm of both sides of the Cobb-Douglas production function to get the following expression:

log(Y(t)) = log(A(t)) + log(K(t)x)+ log(L(t)1-x) (A2)

Using the fact that log(Zb) = b(log(Z)), we can re-write (A2) as follows:

log(Y(t)) = log(A(t)) + x(log(K(t))) + (1-x)(log(L(t))) (A3)

If we lag this expression by one period (i.e. write it in terms of t-1 rather than t) we obtain the following:

log(Y(t-1)) = log(A(t-1)) + x(log(K(t-1))) + (1-x)(log(L(t-1))) (A4)

Subtracting equation (A4) from equation (A3) gives us:

log(Y(t)) - log(Y(t-1)) = log(A(t)) - log(A(t-1)) + x(log(K(t))) - x(log(K(t-1))) + (A5)
(1-x)(log(L(t))) - (1-x)(log(L(t-1)))

It can be shown that the log difference of a variable is approximately equal to its growth rate. That is, for any variable Z(t), we know that logZ(t) - logZ(t-1) @ (Z(t) - Z(t-1))/(Z(t), which is simply the percentage increase (i.e. the growth rate) in Z between t and t-1. Denoting D Z(t)/Z(t) as the growth rate of Z(t), we can write equation (A5) in terms of growth rates:

D Y(t)/Y(t) = D A(t)/A(t) + x( D K(t)/K(t)) + (1-x)( D L(t)/L(t)) (A6)

Equation (A6) is the fundamental growth accounting equation. It tells us that the growth rate of real output ( D Y(t)/Y(t)) is equal to the growth rate of total factor productivity ( D A(t)/A(t)) plus a weighted sum of the growth rates of the capital stock ( D K(t)/K(t)) and the stock of labour ( D L(t)/L(t)).

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