Is it correct that you can always use a lower-wattage light bulb than the lamp is designed for, just not a higher-wattage bulb?
So if the lamp is meant for 40 W bulbs, you can put in a 25 W bulb, just not anything higher than 40 W or it will be a fire hazard?
Any particular complications that can arise from using wattage bulbs that are too weak, other than just dim light?
That’s generally true.
You can run into issues with things like LED bulbs since while they are technically lower in wattage, they concentrate the heat at the base instead of having it spread out around the bulb. This can cause the base and socket to overheat, even when a higher wattage incandescent bulb would have been fine.
I’m sure you can find some oddball bulbs that have similar issues due to their design and where they generate heat.
For most typical lighting situations though, you can safely substitute a lower wattage bulb and the only issue will be that it will not be as bright.
So putting aside the heat issue for the moment, can one replace a 60-watt incandescent bulb with an LED light of 60 watts or less? In other words, if the lumen output of a 60-watt incandescent is the same as a 9-watt LED, can I use a 9, 15, 25, or even a 60-watt LED as its replacement? Of course the greater wattage of the LED than required means more lumen output. But all things being equal, is it OK?
Now the LED heat issue is a problem so even a 9-watt LED (60-watt incandescent equivalent) may increase the heat/fire danger. Is a 15 or 25-watt LED hotter than a 9-watt?
Heat is the main issue in determining how large of a bulb (in terms of wattage) you can put into a particular fixture. So putting aside heat is putting aside your main real-world limiting factor.
As far as the wiring and the socket are concerned, current is current. The wiring doesn’t know or care what type of bulb is in the socket. All the wiring cares about is how much current is flowing. So from that point of view, the wiring won’t catch fire just because you have one bulb type over another.
As far as current is concerned, you’ve got the right basic idea. Anything less is fine. Anything over the maximum for the wire size can cause a fire. The wiring only cares about the actual current, not the lumen equivalency. So while a 9 watt LED might have the same light output as a 60 watt incandescent, from a wiring point of view, the 9 watt LED is equivalent in current draw to a 9 watt incandescent. So yes, if you could find a 60 watt LED, from a wiring point of view only, the wires would be able to handle the current required to drive it.
But then, that’s ignoring heat, and in this case, the heat from a 60 watt LED would most likely melt the socket.
Generally speaking, the higher the wattage, the more heat it puts out. So under most circumstances you’d expect a 25 watt LED to put out more heat than a 15 watt, and the 15 watt would put out more heat than the 9 watt.
Thanks. I have no intention of exceeding the lamp current requirements, nor the heat issue. All my indoor lamps still use incandescent bulbs. I also stocked up on them before several wattages went offline because the early LEDs were hard on my eyes. I’m planning to convert the non-reading fixtures (hallway lights, kitchen lights, closet lights, bathroom lights) to LEDs, and keep my reading lamps with incandescent lights.
The good LED bulbs today have very, very good light quality. However, not only it’s hard to find any E27(standard socket) LED over 15-20W, anything higher would need lots of cooling. A led bulb will fry at 100 C, whereas an incandescent could happily live much hotter.
How are you defining as light quality? My concern is their color rendering such as cool white, white and warm. I don’t want my home to look like a TV studios with all the harshness of bright lights. I want to maintain the warm ambiance that comes from incandescent bulbs. Of course, when I want light so I can find things (a closet) that doesn’t matter so much.
So besides the bulb base size and wattage you need to look at the color temperature. The lower it is the yellower. The TV studio glaring white is most likely 5000K or higher. The typical incandescent IIRC is in the 2500 - 3000K range. I used that when looking for CFL bulbs and recently started seeing it on the LEDs.
A 60W LED bulb is going to weigh a lot more than a 60W incandescent, which may be important for certain fixtures.
The color temperature you want to look for to match incandescent lamps is 2700-2800K. Halogen is more like 3000K, which might be better in bathrooms or the kitchen. Once you get into the 3500-4000K (cool white) range that’s the realm of office fluorescents and metal halides. Anything advertised as daylight is usually around 5000-6500K and looks super blue. Our eyes are more sensitive to blue at low light levels, so daylight looks white because it’s very intense, but a 5500K bulb indoors will look terrible and cold.
Also look at the CRI (color rendering index). 100 is equivalent to an incandescent or daylight at rendering colors as you’d expect and relatively evenly, anything lower is going to be deficient in some way (for instance, fluorescents tend to be worse at rendering reds, but with distinct spikes at orange and a sort of limey green). You generally want a CRI of 90 or better, or you might not be able to tell your black socks from your navy blue ones, among other things. 
There’s some more discussion about the heat issues here: http://boards.straightdope.com/sdmb/showthread.php?p=19883635
And the wiring can carry a current load far above what is needed for any of these bulbs.
The normal lighting circuit in the USA is wired with 14 gauge wire, which is legally rated for 1,600 watts, 26 of those 60-watt lamps. The wiring inside the fixture is at a minimum 18 gauge, which is rated for 840 watts, or 14 of those bulbs.
So the limiting factor is clearly the heat generated by them, not the current.
Doing antique restorations, I’ve rebuilt and/or rewired countless fixtures. Remember heat rises.
An overheated ceiling light could be a real issue. A table lamp with a freestanding exposed bulb, not so much.
But would an LED bulb with an equivalent light output to a 100W traditional bulb be expected to have a similar heat output as well?
I have lampshades made of origami paper, which don’t let much light through. They say max 25W bulb, but I want a lot of light. At the moment I have 15W LED bulbs in there, which are (supposedly) equivalent to 100W incandescent bulbs.
Is that likely to burn my house down?
It depends on the shape of the lamp shade.
It’s designed for 25W where the heat is spread out over the bulb, but your 100W equivalent LED has 15W concentrated at the base of the bulb. If the paper isn’t anywhere near the base, then you’re fine. You’re well under the 25W that the shade is specified for. But if the paper is close to the base, you might have an issue, especially if it restricts airflow.
Something shaped like this might cause an issue, if the paper fits close to the bulb and socket:
Something like this is less likely to be a problem since it is open at the top and will let the heat rise out of it, but still the bottom of it does fit pretty tightly around the base and socket, where all of the heat is generated.
One that is tube-ish in shape and is open on the top and bottom wouldn’t be an issue.
In brand-name LEDs you will normally have a temperature shutoff when the circuit is too hot. Much lower than needed to ignite anything.