# Growth Accounting, capital-output approach

In standard approach to growth accounting, you decompose $$y_t = A_t f(k_t, l_t)$$ into TFP, labor and capital components.

In the capital-output approach, people make the argument that $$A_t$$ can also affect capital as in a solow model, where $$A_t$$ increases, which shifts the production function outward, and $$k_t$$ increases to a new steady state. So $$A_t$$ affects $$k_t$$.

To account for the fact that $$A_t$$ affects $$k_t$$, it is standard in the literature to divide both sides of $$y_t = A_t k_t^{\alpha}, l_t^{1-\alpha}$$ by $$y_t^{\alpha}$$. I can not understand why not divide both sides by say $$A_t$$ to control for it, but instead $$y_t^{\alpha}$$.

• Can you give some references?
– tdm
Nov 4, 2021 at 14:54
• See appendix B of "Understanding Economic Growth in Ghana in Comparative Perspective". Also, Chari uses that in his growth accounting for the USA. Nov 9, 2021 at 15:50

I think that the division by $$y_t^\alpha$$ is for the following specification (where technological progress is labour augmenting).
$$y_t = k_t^\alpha (Al_t)^{1 - \alpha}.$$ Taking logs gives: $$\ln(y_t) = \alpha \ln(k_t) + (1-\alpha) \ln(A_t) + (1-\alpha) \ln(l_t).$$ If we take the derivative with respect to time, we still have a factor $$(1-\alpha)$$ in front of the growth rate of technology $$\dot A/A$$. To get rid of this, we can subtract $$\alpha \ln(y_t)$$ from both sides (i.e. divide the original by $$y_t^\alpha$$): $$(1-\alpha) \ln(y_t) = \alpha \ln (k_t) - \alpha \ln(y_t) + (1-\alpha) \ln(A_t) + (1-\alpha) \ln(l_t)$$ Now divide both sides by $$(1-\alpha)$$: $$\ln(y_t) = \frac{\alpha}{1-\alpha}\ln(k_t/y_t) + \ln(A_t) + \ln(l_t).$$ Then taking the derivative we get: $$\frac{\dot y_t}{y_t} = \frac{\alpha}{1-\alpha} \frac{\dot{(k_t/y_t)}}{(k_t/y_t)} + \frac{\dot A_t}{A_t} + \frac{\dot l_t}{l_t}$$ So the growth rate of technological progress is given by: $$\frac{\dot A_t}{A_t} = \frac{\dot y_t}{y_t} - \frac{\alpha}{1-\alpha} \frac{\dot{(k_t/y_t)}}{(k_t/y_t)} - \frac{\dot l_t}{l_t}$$ If the share of kapital over output, and labour remains constant over time, this means that technological growth is approximately equal to output growth.
• I think it depends on how you define $A$. See also here