If your algorithms get more efficient over time at both small and large scales, and experiments test incremental improvements to architecture or data, then they should get cheaper to run proportionally to algorithmic efficiency of cognitive labor. I think this is better as a first approximation than assuming they're constant, and might hold in practice especially when you can target small-scale algorithmic improvements.

I'm worried that trying to estimate $\sigma$ by looking at wages is subject to lots of noise due to assumptions being violated, which could result in the large discrepancy you see between the two estimates.

One worry: I would guess that Anthropic could derive more output from extra researchers (1.5x/doubling?) than from extra GPUs (1.18x/doubling?), yet it spends more on compute than researchers. In particular I'd guess alpha/beta = 2.5, and wages/r_{research} is around 0.28 (maybe you have better data here). Under Cobb-Douglas and perfect competition these should be equal, but they're off by a factor of 9! I'm not totally sure but I think this would give you strange parameter values in CES as well. This huge gap between output elasticity and where firms are spending their money is strange to me, so I strongly suspect that one of the assumptions is broken rather than $\sigma$ just being some extreme value like -0.10 or 2.58 with large firm fixed effects.

My guess at why: The AI industry is very different than it was in 2012 so it is plausible these firm fixed effects have actually greatly changed over time, which would affect the regression coefficients. Just some examples of possible changes over time:

• Some of the compute growth over time is due to partnerships, e.g. Amazon's $4 billion investment in Anthropic. Maybe these don't reflect buying compute at market prices.
• The value of equity now is hundreds of times higher than in 2017, and employees are compensated mainly in equity.
• The cost of hiring might be primarily in onboarding / management capacity rather than wages; this would hit OpenAI and Anthropic harder since 2021 since they've both grown furiously in that period.
• The whole industry is much larger now and elasticity of substitution might not be constant; if so this is worrying because to predict whether there's a software-only singularity we'll need to extrapolate over more orders of magnitude of growth and the human labor -> AI labor transition.
• Companies could be investing in compute because it increases their revenue (since ChatGPT), or stockpiling compute so they can take advantage of research automation later.

Nevertheless I'm excited about the prospect of estimating $\sigma$ and $\varphi$ and I'm glad this was posted. Are you planning follow-up work, or is there other economic data we could theoretically collect that could give us higher confidence estimates?

(edited to fix numbers, I forgot 2 boxes means +3dB)

dB is logarithmic so a proportional reduction in sound energy will mean subtracting an absolute number of dB, not a percentage reduction in dB.

HouseFresh tested the AirFanta 3Pro https://housefresh.com/airfanta-3pro-review/ at different voltage levels and found:

• 12.6 V: 56.3 dBA, 14 minutes
• 6.54 V: 43.3 dBA, 28 minutes

So basically you subtract 13 dB when halving the CADR. I now realize that if you have two boxes, the sound energy will double (+3dB) and so you'll actually only get -10 dB from running two at half speed. So a more accurate statement for the Airfanta would be that for -15dB noise at the same CADR, you need something like 2.8 purifiers running at 36% speed. It's still definitely possible to markedly lower noise by adding more filter area.

Your box fan CR box data tell a similar story. If logarithmic scaling is accurate, the sound reduction for halving CADR would be ln(1/2)/ln(165/239)*(8 dB) = 15 dB, or 12 dB for maintaining CADR with double the units. It just doesn't have a speed low enough to get these low noise levels (and due to the box fan's low static pressure you might need to add more filters per fan at low speeds).

Airfanta's absolute noise levels are high for a CR box type design but this is a device that retails for 298 CNY = $41 USD in China, runs at high speed, and uses near-HEPA (95%) rather than MERV filters so is to be expected.

FWIW I predict they will be a constant factor harder but improve at similar rates. Any particular benchmarks you think I should look at?

I broadly agree with section 1, and in fact since we published I've been looking into how time horizon varies between domains. Not only is there lots of variance in time horizon, the rate of increase also varies significantly. 

See a preliminary graph plus further observations on LessWrong shortform.

You can get down to 25 dB by running two at half speed. Fan noise is proportional to RPM^5, so 50% speed will mean -15dB noise. The fans just need enough static pressure to maintain close to 50% airflow at 50% speed.

Usage varies-- the top five posts on /r/crboxes all use PC fans. Other guides do too,  and CleanAirKits and Nukit both describe themselves as PC fan CR boxes.

That's a box fan CR box; the better design (and the one linked) uses PC fans which are better optimized for noise. I don't have much first-hand experience with this, but physics suggests that noise from the fan will be proportional to power usage, which is pressure * airflow, if efficiency is constant, and this is roughly consistent with various tests I've found online.

Both further upsizing and better sound isolation would be great. What's the best way to reduce duct noise in practice? Is an 8" flexible duct quieter than a 6" rigid duct or will most of the noise improvement come from oversizing the termination, removing tight bends or installing some kind of silencer device? I might suggest this to a relative.

Isn't particulate what we care about? The purpose of the filters is to get particulate out of the air, and the controlled experiment Jeff did basically measures that. If air mixing is the concern, ceiling fans can mix air far more than required, and you can just measure particulate in several locations anyway.

A pair of CR boxes can also get 350 CFM CADR at the same noise level for less materials cost than either this or the ceiling fan, and also have much less installation cost. E.g. two of this CleanAirKits model on half speed would probably cost <$250 if it were mass-produced. This is the setup in my group house living room and it works great! DIY CR boxes can get to $250/350 CFM right now.

The key is having enough filter area to make the static pressure and thus power and noise minimal-- the scaling works out such that every doubling of filter area at a given CADR decreases noise by 4.5 dB, assuming noise is proportional to power and pressure goes as (face velocity)^1.5, which are common rules of thumb. I'd guess that the pair of CR boxes has 5x more filter area, so an 11dB advantage for the closet sound isolation to make up. MERV filters also get slightly higher efficiency when the face velocity is slower.

I have used inline fans for other purposes and even the air passing through a 6" duct generates some noise and adds static pressure. With a CR box you're doing the minimal work necessary to filter air.

Standard HVAC parts do have many advantages though. The aesthetics are unmatched and all parts are likely to be available, and they're very durable.