# 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

## 1 Answer

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 don't think that it is for this specification only...see appendix B of this paper, "Understanding Economic Growth in Ghana in Comparative Perspective". There are many like that in the literature. The argument is that TFP can also affect capital, but not sure why divide by y^alpha. In the literature, it is not model specific, like you suggest. Nov 9, 2021 at 15:49
• I think it depends on how you define $A$. See also here
– tdm
Nov 10, 2021 at 8:19
• Thanks. Nice article. As mentioned in there, the reason why they divide by y^alpha is that they wanna scale up the importance of TFP in the accounting process. I think that answers my question. I have to look for why they wanna scale TFP by a factor of 1/(1-alpha) and not some other factor though. Nov 10, 2021 at 12:49