Here's a quick tip if you have some fractions that you need to simplify. Just cancel any digits in common between the numerator and denominator.
Here are a few examples...
I was one of the worst French students in my family. My brother is actually quite fluent, having spent some time in New Caledonia, and acquiring a very amusing New Caledonian accent. My mother has spent more time in France than I ever did; and my father studied it for his leaving certificate. I dropped it at the end of year 10 (4th form as it was known at the time) before I started my pipe-organ building apprenticeship. I wrote my end-of-year French exam left-handed just to practice — that’s how little I cared about French. Sorry, Ms Kviz, it wasn’t your teaching.
It was doomed to end badly. I translated two sentences that day in 1990 and got stuck on the third.
But I promised to finish it later.
In my family we have an alphabetic Christmas tradition — each year the presents begin with the next letter of the alphabet, which for the year 2020 was the letter “T”. So I finished the translation (30 years after I promised to do it, and with a lot of help from my daughter, who started French in kindergarten) and gave it to her for Christmas.
At the end of last year as I was wrapping up the first draft, I found that a lot of the women in my life started asking about it. Heidi was a part of their childhood, and they would like to know how it ends, so I’ve decided to publish it. I had some delightful watercolours done — based on the original that I was working from.
It is available for pre-order for March 1st 2021: https://www.amazon.com/dp/B08WH6774S
Landfill is an interesting product. The price fluctuates far enough that it can fluctuate from positive to negative and back again. If someone is building a tunnel, you can probably get paid to take landfill. If someone is building an island, you can probably get paid for supplying landfill. There are projects that can only go ahead at certain times: the Hornsby quarry remediation project was only viable because the north connex was being built.
That's slightly obscure, and there's not much you can do with landfill other than, well, fill land. But we are about to hit a very unusual combination of circumstances which will create some head-scratching economics.
Solar panels are getting cheaper. The learning rate for solar energy is (probably) the steepest of any energy technology yet deployed. So we can be pretty much assured that within the next decade we'll have enough solar power to power anything we want (as long as it is day time). We'll overprovision our houses and factories (and offices if they still exist) with generation capacity so that in the morning or evening when the sun is hitting at an oblique angle, we still have enough for our needs.
That means that at peak times, we'll have more than enough solar energy. What do we do with that surplus energy?
Actually, it's slightly worse than that, because there are some (non-renewable) power plants that can't fully idle. A nuclear reactor can't fully turn off for a few minutes in the middle of the day. Neither can a coal power plant. So they will be running at a time when nobody has a need for power. That may be OK -- if you make a lot of money generating power overnight, you might be prepared to pay some small penalties during the day for overproduction if that's what it takes to keep the system running.
Which means that someone, somewhere is going to get paid to create a dummy electrical load. We've never (as far as I can tell) seen "grid stability" payments for using electricity before. But it's the inevitable conclusion of where we are the moment: discounted rates for consumers who can change their requirements to help with grid stability exist already; there's no reason that this wouldn't still be necessary when the price of electricity hits zero.
Therfore, there will be times of the day (in certain weather conditions) when some companies will get paid to use electricity.
The kinds of consumers that will get paid are likely to be doing something that is energy inefficient (obviously), and not particularly capital intensive (since their equipment will be idle a lot of the time), and making something that can be stored indefinitely (because you definitely can't do just-in-time manufacturing if you are waiting for times when the price for electricity goes negative). But quite a lot of products could be made to fit that.
A possible product that I think is likely to play out this way is water desalination. Desalinated water is very expensive because of the embedded energy cost: it's often described as "bottled electricity". Currently we tend to use reverse osmosis because it's more energy efficient; but if you don't have to worry about energy costs, you might choose instead to boil water and distil it: since the capital costs are cheaper.
Now things get even weirder. Let's say you are getting paid 0.5c per kWH (a not unreasonable price) that you use. You might find that you can sell the water for -0.01c per kL. Yes, you can make a profit: not just giving it away, but you could still be profitable while paying someone to consume it.
If this all plays out, we might have water and electricity (at times) having a negative price. These are common inputs to other processes, so it doesn't end there: there might be a business that requires water and electricity occasionally, but the input costs normally make the business unviable. They in turn might be able to make some money selling the end product at a negative price if their inputs have a negative price.
If this sounds all a bit unbelievable, remember that up until very recently, many economists thought that it was impossible to have negative interest rates... right up until the moment when they started happening. We already have examples of products that oscillate between positive and negative prices, it's just that they were never fundamental inputs to other processes. My prediction is that by 2030 we will have a small part of the economy ticking away profitably making negative price products.
(Post scriptum: now for the completely weird: how do negative price products interact with negative interest rates?)