Currently, I have a 400W PSMH Westinghouse lamp made in China, and I also have a GE M400 in 400W HPS, which are obviously two different lamp technologies. Recently, I was wondering what would happen if I ran the 400W PSMH lamp on the 400W HPS ballast. The arc voltage drop is higher than that of a 400W HPS lamp, so how would this affect the ballast? Would it overload the ballast?

Going by reference ballast characteristics (not exactly realistic, but good enough), the 400W HPS needs 38.3 ohms at 220V while the 400W MH needs 45 ohms at 220V. That is a 16% difference, so not insignificant.

According to my Desmos calculator which has a history of being semi-correct, running a 400W MH lamp on 38.3 ohms at 220V will run it at almost 4 amps, which is a 23% lamp overdrive, assuming that the lamp voltage doesn't significantly decrease from the normal 135V (not exactly realistic, but good enough). Very significant lamp overdrive, but probably fine for short periods or with a lamp you don't care about.

This would mean that the ballast meant for ballasting 400W lamps is now ballasting a lamp that is operating at more than 400W (probably around 500W if I were to take a wild guess). The secondary coil would probably be fine since it would be carrying less current than normal, but the primary will probably carry more current than normal due to the increased secondary wattage. If you choose to do this, just monitor the temperature and make sure it doesn't get too hot I guess. Long-term it will certainly reduce ballast lifespan.

I second Multisubject's answer

There are PSMH lamps for the European market rated for this exact use (400W HPS ballast), but there the ballast is just a choke so there is no primary winding to overload

500W is a significant overload, so unless the ballast is running in a significantly reduced ambient temperature (able to compensate for ~20W ish additional loss), this will probably lead to overheating of the ballast

A ballast of this size has high thermal capacity, and may take well over an hour to stabilize at steady state temperature for measurement

Currently, I have a 400W PSMH Westinghouse lamp made in China, and I also have a GE M400 in 400W HPS, which are obviously two different lamp technologies. Recently, I was wondering what would happen if I ran the 400W PSMH lamp on the 400W HPS ballast. The arc voltage drop is higher than that of a 400W HPS lamp, so how would this affect the ballast? Would it overload the ballast?

Is your 400W ballast of HX or CWA topology?

CWA

So it goes down to some US-specific ballast behavior.

With regular HPS Euro series chokes or HX transformers, 3.25A mercury compatible MH lamp will get somewhat overloaded, but not up to 4A due to higher arc voltage, guess something like ~3.7A current. Ballast itself of course will run colder because of lower than nameplate current.

CWA ballast though has a chance to push the lamp a bit further due to current-stabilizing behavior. BUT I heard that HPS CWAs load curve is specially tamed not to overdrive aging HPS lamps with rising working voltage and not to force them into cycling. So it may be not that bad, actually.

So, would running a 400W PSMH lamp on a 400W HPS ballast cause the ballast to overheat? I'm not worried about the lamp; it's a cheap Chinese-made Westinghouse lamp. I'm more concerned about damaging the ballast.

@NeXe Lights
The ballast would be working harder than it is meant to. Whether or not this will happen to the point of it overheating in uncertain.

It would definitely not cause immediate damage to the ballast, so monitoring the ballasts temperature during operation is the best way to proceed. If the ballast starts getting too hot, cut the power. You almost certainly won't cause any damage to the ballast if you maintain reasonable temperatures.

Leaving this to run continuously at high temperatures for days/weeks/months on the other hand, that would probably shorten the lifespan of the ballast, possibly causing failure.

If you plan on doing this a lot or if you notice the ballast overheating. I'd get one of those HPS CMH retrofit lamps that are rated for operation on HPS ballasts.