# How is the Solow residual measured?

In the Solow model, we have the Solow residual often referred to as the level of technology A.

More particularly : \begin{equation} Y(t) = [K(t)]^{\alpha} [A(t)L(t)]^{1-\alpha} \end{equation}

Here it is defined as "is the portion of output not explained by the amount of inputs used in production" though there's an explanation of its fluctuations, I still do not understand how it is calculated.

This paper from the National Bank of Belgium does leave me a bit confused because several measurements are said to be used but it sometimes sounds as if one took the total output for a year and substracted what wasn't explained directly by labour and capital, to obtain the residual?

I'm not really sure how it is measured and where said measurements even come from,

Can someone point me to an answer?

Nate

Below please find a portion of a lecture slide a professor of mine used last year. Please note that $\gamma_{\tilde{y}}$ denotes per-capita output growth, $\gamma_{\tilde{k}}$ denotes per-capita capital growth and $\alpha(t)$ denotes the income share of capital. $R(t)$ - the Solow residual - can then be easily computed.

The production function you have stated was the one we used for this example. So it can be applied here directly. Source: "Growth and Business Cycles" lecture by Professor Gerhard Sorger, University of Vienna

• thanks for your input but the same problem arises for me, we're calculating the value ex post once we already know the value of the output. α seems to be calculated empirically while the residual value is poorly explained in my opinion, because it always relies on already knowing the ouput and α. – Nathan Furnal Mar 12 '18 at 16:30
• @DiogenicOrder Yes, you are right in saying that it is calculated ex post. But calculating the residual is an empirical exercise or else you could use your assumed $\gamma_{A}$ for a theoretical model. This holds for any growth rate. If you are looking for instantaneous economic growth rate updates please see “GDP nowcasting”. If you are looking for ways to model technological growth per se, please see some endogenous growth theory models. – WillTheGeek Mar 12 '18 at 16:38
• @ WillTheGeek Great ! I'll look it up, any resources in mind like a book or a course I could read? Thank you ! – Nathan Furnal Mar 12 '18 at 16:42
• @DiogenicOrder I am not sure at what level of economics you are but: 1. On introductory growth theories I can recommend Acemoglu's "Introduction to Modern Economic Growth" or Romer's "Advanced Macroeconomics". 2. On GDP nowcasting, just have a look around Google Scholar searching for the term. Several papers should pop up. P.S. Please do me the favor and mark the question as answered, so other economics.stackexchange users will have an easier time finding an answer to their Solow residual question. – WillTheGeek Mar 12 '18 at 18:26
• @ WillTheGeek At undergrad level, that question has just been bugging me for some time. Thanks for your answers, I'll do so. Have a good one! – Nathan Furnal Mar 13 '18 at 12:42