What do you mean "you were using"? Gravitational potential energy at Earth at sea level is 9,81 * ChangeInAltitude * mass. 35,5 m/s * 9,81 * 20000 = 7MJ/s = 7MW. If you "were using" 2,4MW then you were only climbing at 12,2m/s meaning your entire trip takes 41 days - over a month. Which means that your elevator has laughably worthless throughput. And 20k kg climber requires a massive elevator massing millions of tonnes *with* unobtanium. So you're proposing to launch millions of tonnes of unobtanium to GEO in order to send a fraction of 20tonnes up once every 41 days? Good luck with that.
You could expect 60% efficiency
That's exceedingly optimistic even for monochromatic light (which I see we're back to discussing). Have you ever priced the sort of Spectrolab cells you're proposing here? And anyway the highest monochromatic conversion rate ever recorded - lab scale - was 53%.
Remember that PV efficiency goes up as the light gets brighter
Only when you can keep the cells cooled to the same ambient temperature (and it's only a relatively small gain). How exactly do you propose to ditch megawatts of waste heat up there? Heat is a killer to solar cell efficiency. And several megawatts shining on a relatively small area is otherwise known as "vaporizing it".
No comments about the 0,1%-ish efficiency of the sorts of lasers that actually have the coherency and power to beam over such distances, I see. Even over the distances of your "in-orbit" lasers, of which apparently you want there to be hundreds of thousands if you want to ensure that there's one close to the tower at all altitudes at all points in time. Hundreds of thousands of multi-megawatt lasers each consuming a gigawatt or so of power. In order to launch a fraction of 20 tonnes to GEO once every 41 days. Great strategy.
Economically the construction cost will be huge, but once you have one you can build more relatively cheaply because it costs very little to get mass into orbit.
There is nothing "cheap" about what you're proposing. Your capital costs are nonsensically high, and you have to pay interest on capital costs if you want to live in the real world, and interest accrues interest. You will never, ever reach an economically valid argument for it. And for what gain? If you're turning $0,08/kWh industrial-rate grid electricity into climbing power at 0,05% efficiency then you're paying $160/kg to get to orbit, several times the price to orbit of what's possible with a rocket if it can be made reusable with minimal turnaround costs between flights (as mentioned earlier, the Shuttle's propellant cost to orbit was only $80/kg, most of that in the SRBs, which aren't the cheapest of propellants). And of course it's not even close to a Lofstrom loop, which can be made without unobtanium and deliver payloads at an energy cost to orbit of about $1,60/kg, with present tech.
Speed isn't a huge problem if your cable can support multiple climbers.
So you want to make your cable even bigger, heavier, and more expensive. How many times more expensive do you want to make it? 5 times? 10 times? 100 times? Why not just say that your cable is going to be the mass of the moon's worth of unobtanium while you're at it?
And again, we're only talking about the most basic of problems with space elevators here, let alone actually getting into the countless engineering problems, some of which have no known solutions, and none of which you really have a mass safety margin to properly address. The resonance issues are some of my favorite ones: from the climbers, from the atmosphere, from the sun and from the moon. You have a giant cable which has basically zero ability to damp itself, and no mass leeway to install any sort of damping system of the sort of magnitude needed to counter oscillations. On top of the fact that even made of unobtanium it's an ultrathin structure that can barely support itself and has to be able to withstand hypersonic impacts of microscopic debris and long-term exposure to the Van Allen belts at the time time, while in the atmosphere it's going to face wind loading (they call it a ribbon for a reason, and ribbons *blow*), potentially many times higher than that of climbers, potential icing, certain wetting, lighting (which even if the cable itself isn't conductive, the water on it will make for an easier ground path than the air), upper atmospheric (sprite) lightning as well, oxidation (no mass margin for protective coatings), and on and on down the line.
The space elevator concept needs to be consigned to the dustbin. It was a neat thought experiment for a while until the real-world hit. And now we have actually potentially workable structures (the actively suspended ones) that supercede it in every measure, so there's no point to it.