I'm going to go over the causes for hair loss FIRST and then talk about your dog specifically as well as making some comments while discussing other causes.
Unfortunately, there are many conditions that can cause a dog to experience hair loss including ringworm, mange, hypothyroidism, cushing disease and allergies. To me it sounds like a yeast infection of the skin possibly caused by an allergy, but I'll go over other causes.
Ringworm can have a scaly reddish rash around the edges of the bare patches though there can be a rash in the bare area as well. Any over the counter antifungal cream for ringworm will work on this. The cream should be applied to an area slightly larger and for about a week after it appears to have cleared up. This is easily diagnosed by your vet.
There are two types of mange and mange is actually caused by mites. Usually hair loss starts in specific areas of the body before spreading to other areas. Sarcoptic mange tends to start lower on the body and is contagious while demodectic is not contagious and tends to start on the ear margins and face. This usually strikes dogs under a year of age and dogs with a compromised immune system.
Allergies are a frequent cause of scratching and hair loss and an be caused by many different things. Treating any problem without knowing for sure what it is causes a delay in the treatment of the real problem if your first guess is wrong. I really suggest you have your Vet examine the areas and diagnose this. Below are websites that will go into more detail on these conditions.
Here is a website on Ringworm:
Here is two on Mange.
Hypothyroidism and cushing on these pages:
You can read up on allergies here:
If you suspect an allergy condition is to blame, you can give your dog Benadryl up to 2mg per pound every 8 hours to help with symptoms..
There might be a secondary skin condition in addition to the allergy such as staph dermatitis or yeast dermatitis. This is what I believe is the likely cause of what you are seeing. Staph usually occurs on the lower regions of your pet and tends to have small pimple type bumps. Shampoo containing Chlorhexiderm and/or Oatmeal can help with this condition though it does not cure the allergy. Yeast typically shows as a greasy area that has a sweet musty odor. Sometimes the skin can become inflamed, darker and thickened due to itching. Yeast likes areas such as between toes, armpits and ears. Selsun Blue Shampoo can help with Yeast dermatitis. When shampooing, lather and leave on 15 minutes before rinsing. These shampoos are not meant to be a cure, just a relief until your pet can see a Vet.
There is also a condition called alopecia which is basically hair loss. You can read all about it in this article:
Some treatment methods for this are discussed on the following site. You might want to discuss this possibility with your vet.
Since this is in the lower region of your dog, it might be sarcoptic mange mites as well, but it isn't unusual to see allergic reactions in this area either. I do think you need to have your dog seen by your vet and have a skin scraping done to eliminate mites as the cause. Before you get into the vets you can try the selsun blue shampoo and even some miconazole cream (ask your pharmacist for this) on the areas. It won't hurt and if this is a yeast dermatitis, you may start seeing improvement before your dog is seen.
I'd still have your dog seen as there is likely to be an underlying allergy or other condition causing the itching that lead to the skin dermatitis and hair loss.
I hope this information is helpful.
The problem with skin issues is that any condition that causes a dog to be itchy such as fleas, allergies and mites, can lead to scratching. The dog scratches the skin and this damages the skin's immune system and the yeast or staph normally present on all dogs is allowed to multiply unchecked. The yeast and staph when multiplying unchecked leads to additional itchiness and scratching which in turn leads to hair loss and red skin patches such as you describe.
Since many times there is an underlying cause for the dermatitis, I feel it is appropriate to discuss the various underlying causes as well as what I feel is probably the problem. I did state that I thought a skin dermatitis such as yeast probably what you are seeing and suggested using the selsun blue shampoo and miconazole cream to help clear it.
I do have to recommend a vet visit though as often a skin dermatitis is secondary to another problem such as mites or allergy. You could try the shampoo and miconazole cream and see if there is improvement. You might check the food if a switch was made before you noticed the itching and switch back to the old food, thus possibly eliminating the allergen if an allergy started the problem. You could use the benadryl as well to help with itchiness.
Unless a skin scraping comes back positive for mites, even your vet is likely just to treat the skin dermatitis first with shampoos, possibly antibiotics and steroids to give the skin time to heal and then see if it occurs again before considering allergy testing. .