For guarding a typical flock of chickens they eat a lot. After using dogs to protect sheep and goat herds of about 50 animals with one dog per herd, the dogs were deemed to expensive and swapped out for donkeys. Dogs are not cheap. You also have occasional vet bills with them as well. Herding dogs will also get tired of herding birds and ignore them looking for more stimulating activities. Bird dogs most certainly tire of molesting the birds.
For a typical flock, half a dozen birds, sure, if you are looking at dollars and cents. It is a personal choice though and some don't look at every cent spent. If they were, they likely wouldn't be raising chickens in the first place. Most have them for enjoyment and good quality eggs. Most would love to give their birds the ability to free range and if you live in the right area an LGD can make that possible. One can always find cheaper quality food, take the dog to the shot clinics (much cheaper), and do your own worming, etc.
Most don't have the ability to put up electric fencing, or even hardwire cloth fencing. You mention fire hazard, another thing to consider when electrifying a fence. An electric fence wouldn't work where I'm at for several reasons and I'm not willing to fence three acres like a fortress with hardwire cloth. I'm not protecting a half dozen birds, but several dozen and birds that cost me, initially, a pretty penny. They are my hobby, not an income producer. Because of my choice, my girls can free range all they want, saving me $ in feed. My dog is well worth her small expense and because of her I can go out at night and not worry about me being surprised by a bear, bobcat, coyote, etc. She is always on guard or has an eye or ear open around here. She works even when a fence isn't. She didn't cost hundreds of dollars, doesn't eat much at all, and I have had no major medical problems with her. I spend more on the chickens than I ever spent on her total. For me, and I'm sure others, an LGD is the better option, because the fence would be the waste of money. Meaning, it would likely be shorted out often around here, not give me the ability to free range and bite my grandkids when they get near. Don't much feel like driving post in rock either. Nope, not for me.
And I sure wouldn't want to wait for a bird dog to tire of chasing the birds. Could lose your whole flock before it tires. If you already have the dog before the flock, sure, make the effort to train it. If not, and you are looking for a guardian dog, get an appropriate breed.
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