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I would like to quantify the importance of computing power for economic aggregate output in an economy in a "simple" GDP-to-installed-CPU ratio.

The idea is to calculate a measure similar to GDP-to-capital.

The challenge lies in quantifying installed CPU power in an economy. That is, to define an adequate (and feasible) measure of installed computational facilities and have an idea how to measure this.

There have been objections to a previous version of this question on the basis that this is difficult. It might, in fact, be.

Any suggestions on parts of this problem are highly welcome.

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    $\begingroup$ GHz is a unit for frequency; 10^6 cycles per second. A “GHz hour” (or MHz hour) makes no physical sense. Go you mean gigawatt-hour? $\endgroup$ – Brian Romanchuk Apr 8 '18 at 19:35
  • $\begingroup$ Also which country you are talking about is probably relevant. $\endgroup$ – denesp Apr 8 '18 at 20:17
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    $\begingroup$ This would probably be very hard to quantify, because you rarely use computing power to directly produce something. $\endgroup$ – denesp Apr 8 '18 at 22:15
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    $\begingroup$ The concept of a GHz-hour (or MHz-hour) is very unusual but apparently it is not completely unknown. There are some companies that charge by the GHz-hour, e.g. Pixel Plow charges \$0.005–\$0.006 per GHz-hour for its services. Unfortunately, this concept seems to be very unusual even in computing circles. It is certainly completely unknown in economics, so I think it safe to say that the answer to your question is "No". $\endgroup$ – Kenny LJ Apr 9 '18 at 9:34
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    $\begingroup$ The better unit is “flops” (floating point operations per second); Hertz has a well defined meaning in electrical engineering and physics. The answer is that there is no good relationship; you can run a regression between computing power and GDP, but the regression coefficient will change. For example, there were exactly zero digital computing cycles per unit of GDP pre-World War II... $\endgroup$ – Brian Romanchuk Apr 9 '18 at 13:00
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I have electrical engineering training, and the focus on CPUs creates a technical issue. Before the early 1990s (when I did my degree), CPUs would have largely been only embedded in what are recognisably digital computers. Electronics would otherwise be implemented with simpler digital/analog circuits. Currently, the path of least resistance is to just use a CPU for practically everything. For example, cars have plenty of embedded CPUs, whereas back in the 1970s, many might have been almost entirely analog circuits (the radio being the only possible exception).

So any measure of CPUs per unit of GDP should be exploding higher. This is just a reflection of electrical engineers’ design preferences.

If we just try to confine ourselves to CPUs in digital computers, we could try to compute some normalised computing capacity. (This raises problems: do we include phones? Game consoles? Tablets?)

The next problem is: so what? My computer CPUs have been getting faster over the decades, but at the same time, the bloated operating system chews up more of the resources. I end up doing roughly the same tasks in the same amount of time. Even without having any data, knowing what we know about computing power, my experiences are obviously be widely replicated. That is, real GDP is growing at around 2% per year in the developed countries, but installed CPU capacity has to be growing way faster than that, even if we take a narrow definition of “computer”.

Realistically, you might need to identify some industries that might show a correlation between their growth and CPU usage. It’s hard to see how adding server farms can increase the production of restaurants and barber shops.

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  • $\begingroup$ You can also see it like this: with computing moving away from the desktop computer and towards intelligent objects, phones, cars, ..., including all these CPUs would be desirable. As you point out, this will show that computation capacity per $ of GDP has been exploding. But that's the whole point: I wanna show how this is used as an input factor in the economy. This includes CPUs in displays, phones, but also server farms. $\endgroup$ – Murphy Apr 15 '18 at 6:38
  • $\begingroup$ @Murphy - Well, that would suggest that computing capacity is becoming spectacularly less useful over time, since we appear to need way more computing power per unit of GDP. Not sure why that would be useful? $\endgroup$ – Brian Romanchuk Apr 15 '18 at 20:34
  • $\begingroup$ It is not entirely correct to infer from increased usage of an input factor that its becoming useless. It is just getting cheaper and possibly substitutes for sth else. I appreciate the effort, but I think your answer does not directly relate to the question. $\endgroup$ – Murphy Apr 16 '18 at 20:22
  • $\begingroup$ But the point about the design preferences is well taken. It probably means that no measure of CPU capacity is available. Unfortunately, this is not something to say with certainty. $\endgroup$ – Murphy Apr 16 '18 at 20:31
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Here is how an economist would approach the question with a simple model. Of course, a few assumptions are needed. Suppose aggregate production in year $t$ can be thought of as a single firm producing all output ($Y$, which is GDP) using labor ($L$), computers ($C$) and other types of capital ($K$).

$Y_{t}=f\left(K_{t},L_{t},C_{t}\right)$

Let's also make the simplifying assumption that computing speed is captured by the depreciation process of computers. If computing speed is important, then computers depreciate faster as newer, faster models come in. As a consequence, you need to invest in computers more frequently as the economy grows in order to keep the same level of output. Therefore, I argue, a sensible measure of the importance of computing power is given by the growth rate of the economy ($g$) plus depreciation rate ($\delta$), times the elasticity of output with respect to computers (that is, how many computers you need to produce an additional unit of output)

$\left(g+\delta\right)\cdot\left(\frac{\partial Y_{t}}{\partial C_{t}}\cdot\frac{C_{t}}{Y_{t}}\right)=\left(g+\delta\right)\cdot\frac{\partial\log Y_{t}}{\partial\log C_{t}}$

We can use neoclassical growth theory to obtain this measure from the data. Suppose that the economy is on what neoclassical economists call a "balanced growth path". That is: output and inputs all grow at a constant rate $g$:

$\frac{d\log Y_{t}}{dt}=\frac{d\log K_{t}}{dt}=\frac{d\log L_{t}}{dt}=\frac{d\log C_{t}}{dt}=g$

Then the ratio of yearly investment in computers as a percent of the stock of computers needs to be equal to $g+\delta$ in order for growth to remain "balanced"

$\frac{I}{C}=g+\delta$

multiply both sides by the nominal value of the computer stock ($rC$) to nominal GDP ($PY$) where $r$ is the rental rate of computers and $p$ is the GDP deflator. This gives you the following equation for the yearly nominal investment in computers, as a percentage of GDP:

$\frac{rI}{pY}=\left(g+\delta\right)\frac{rC}{pY}$

now, in equilibrium, the share of capital in total compensation $\frac{rC}{pY}$ is equal to the output/capital elasticity ($\frac{\partial\log Y}{\partial\log C}$). This means that you can rewrite the equation above as:

$\frac{rI}{pY}=\left(g+\delta\right)\frac{\partial\log Y}{\partial\log C}$

In other words, a simple neoclassical growth model suggests capital investment in computers as a percent of GDP as a measure of the "importance" of computing power in the economy.

The data you need to compute this ratio (for different countries and sectors) can be found on the website of the OECD or the EUKLEMS consortium. You'll need to search a bit for it (computers are a subgroup of "ICT capital").

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Without worrying about the precise numerical outputs, you could consider electronics purchases (perhaps subdivided by consumers, businesses and government) as a proxy for CPU consumption. (You could probably come up with some method to roughly indicate CPU power per dollar in different categories by manually figuring this out using whatever best data can substantiate that. E.g.: CPU power per dollar spent on a selection of commonly purchased TVs, laptops, phones, servers, etc.)

Then, just do the analysis using electronics purchases, and don't treat the numerical outputs as important for precision, but instead for a general indication of direction of effect, or the relative size of the effect, e.g. for consumer, business and government sector electronics purchases.

Explain some rationale legitimizing the use of the proxy for the question of interest, then come up with an appropriate term to reflect the question of interest in consideration of what data in fact underlies it.

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